Knights and Knaves: A Logical Adventure
by The Texas Wyrm
Summary: Using a forbidden template, the inhabitants of Gensokyo now find themselves to either be unable to lie, or unable to tell the truth. Follow Remilia Scarlet as she navigates through the logical morass laid down by the Youkai Sage.  Thinking required.
1. Remilia Awakens as a Knight

_Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

"I'll take this one."

Rinnosuke Morichika, the Unmoving Used Goods Seller and proprietor of Kourindou, looked up at one Yukari Yakumo, Youkai Sage of Boundaries, her voice cutting through the normal stillness of Kourindou. She held up a small, orange, ragged paperback book that loudly proclaimed its own title in one of those western scripts as _What is the Name of This Book?_ with a subtitle 'The Riddle of Dracula and Other Logical Puzzles'.

It was a strange title, and the book's author Raymond Smullyan was unfamiliar to Rinnosuke Morichika, but Rinnosuke had determined that the odd book was exactly what it said on the tin, for the most part. It was a book of logical puzzles, with a small treatise on the subject of the nature of proof.

Rinnosuke had not known Yukari to be at all interested in logic in the time he'd known her. He supposed this would be an improvement, since exercising her logic may calm her down.

She even paid full price for it.

—/—

As Yukari Yakumo gapped into Mayohiga, clutching the strangely-titled book and wearing a huge fox-like smile on her face. Ran saw that smile and immediately knew that her master was up to no good.

"Yukari-sama, you're getting one of your ideas again, aren't you?" Ran said, more like a statement than the question it was dressed up as.

"Oh, yes. Amusing ideas. Tantalizing ideas. Ideas that will twist the heads of our friends into little knots and drive them to frustration," she said with a chuckle.

"Master, you are already known more as the 'Troll of Gensokyo' than the 'Youkai Sage of the Boundary.' May I suggest... you know... behaving a bit more sage-like?"

Yukari's smile did not slip as she turned toward her shikigami. "Tell me, dear Ran, why is someone called a sage?"

"Because they have demonstrated deep experience, sound judgement, and impeccable wisdom," Ran replied, as if from a dictionary. "None of which you've displayed of late. Playing pranks on people demonstrates none of those. _Fairies_ play pranks, and they're dumb as rocks."

"Oh, ye of little faith. This will be cool, I promise."

"Your words do not inspire much confidence."

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 1:_

_In Which Remilia Scarlet Awakens as a Knight_

* * *

_Months Later_

Evening fell upon the Scarlet Devil Mansion, situated on an island in the middle of the Misty Lake. It was time for one Remilia Scarlet to rouse herself into wakefulness.

To all appearances, today was as any other. Remilia waking in the night in her pajamas. Shuffling out of bed, yawning and rubbing her eyes of the sleep-boogers they'd collected. "Mmm... Woke early today. Perhaps Sakuya doesn't even know it."

A knock at the door.

"M'lady, the midnight meal," came Sakuya's voice.

"'Midnight'?" gasped Remilia in surprise. "Did I really sleep that late?"

"Indeed you did, m'lady. Do you wish to have your meal in bed?"

Even though she had just gotten out of bed, she did indeed wish to take her meal in bed. "Yes," she immediately and truthfully answered, then stopped, puzzled. It wouldn't do for the lady to be seen having breakfast, or the midnight meal, in bed where crumbs would get into the sheets. "But I must decline," she added, "for there are too many problems for breakfast in bed, especially for cleaning it up."

"Indeed, there is a problem. M'lady does need concern herself about how she takes her meals. It's an embarrassment."

A deadly chill of a silence descended upon the immediate area. "Care to say that again, Sakuya?"

"I-I said, 'There is no problem, and the lady of the house need not be embarrassed taking her meal anywhere she wants,'" came Sakuya's stuttering response.

"That's not what I heard, but I accept your correction," Remilia replied, and went to her bed, a bit steamed. She thought about her words. Why hadn't she said, 'That's what I thought!' in a sarcastic, reproachful tone? It was on her mind, but instead Remilia delivered a completely candid response.

Sakuya Izayoi opened the door with a sickly look on her face and a forced smile. The perfect maid had made her first slip-up, and it was a doozy. "M'lady, this is your midnight meal," she said as she set the tray down with no small nervousness in front of Remilia.

Remilia stared at it.

Buttered toast and jam.

Tea and juice.

Very light. Not much meat or dairy.

Was it brunch? No, Sakuya said it was the midnight meal. A full lunch. It should be heavier. Remilia was about to address this when she spied the time on her mechanical clock: just after sunset.

The mistress's anger deepened. "Sakuya, I thought you said this was the midnight meal! This is a usual breakfast, and the clock says that it should be around the time I should have breakfast anyway!"

"Th-The clock isn't correct!"

Remilia, her suspicions raised, turned to stare out the window. She noticed the dusk not quite ended, and the particular placement of the stars indicated that it was indeed early evening. "Sakuya..." she murmured dangerously, "...the stars and the last fading traces of twilight indicate that the clock is accurate. This is early evening. This is breakfast. Why did you lie and say this was the midnight meal?"

"I-It was a j-joke!" came her pitiful excuse from her pitiful face.

Remilia set firmly down the butter knife she'd been idly playing with. "A joke? It's not very funny, Sakuya."

"Yes, it is!" contradicted Sakuya, increasingly distraught.

"Don't you dare contradict me, Sakuya! If I say something is not funny, it isn't funny!"

Sakuya cringed in mortification, and feebly tried to disengage from that conversation. "I hope you don't enjoy your meal," she murmured. At realizing the words that escaped her mouth, as well as Remilia's furious look, Sakuya tried to recover, "I-I mean, I hope you choke and die on it! I mean—"

"ENOUGH!" barked Remilia. "It's woefully apparent that you resent your employment here!"

"Th-Th-That's true, m'lady" blurted Sakuya, flustered at her mouth seeming to spiral out of control. "I mean, I _do_ resent being under you! I mean, I'd rather die than—"

"SHUT UP!"

Sakuya did as she was told with only a whimper. Tears were spilling from her eyes and took on a miserable face.

Remilia sighed. It was the end of an era. Even being the evil Queen of the Night, Sakuya was the one human she felt closest to, and wanted to see happy. It was obvious to her now that Sakuya was unhappy and rebellious. Perhaps she needed to let her go.

"You're fired, Sakuya."

Sakuya broke down, openly sobbing. "M'lady, I am grateful that you have fired me!" she wailed.

This drove Remilia absolutely livid. "Fine! If you're so happy about it, then I'm lucky I discovered this now rather than— _ack!_ What?"

The 'ack' was yelp of surprise as a small, orange, ragged paperback book popped out of nowhere to whack her on the back of the head.

Remilia snatched it up and brandished it hotly. "What the hell is with this book?"

Sakuya said nothing. Her mouth wasn't serving her very well thus far, so it was probably a good choice.

Remilia finally deigned to look at it, and blinked at what she saw. "'What is the Name of This Book?' Strange name for a book. It's like the author couldn't commit to a title. Like the Dracula on the cover, though." She plucked contemplatively at the bright pink Post-It note in the book, serving as a persistent bookmark. "There's a bookmark, too. Someone wants is to read this, I think. Let's see what it has to say..."

And she read.

She read about a mystical island in the unknown parts of the world. It was a land divided into two kinds of inhabitants: scrupulous truth tellers and unrepenting liars, called knights and knaves, respectively. The book used that premise as the basis for various puzzles.

Then she thought of Sakuya's behavior. It almost seemed to Remilia that she was saying something other than what she wanted to say. As if she wanted to tell her something different, but the words tumbled free without her consent. Could Remilia have a similar affliction.

She tried to formulate a lie to tell. An untruth that would prove that the book was nonsense. But she could not. No matter how she tried, a sentence she knew to be untrue could not pass her lips: 'two plus two equals five,' 'I am a teapot,' and many others. No amount of force could get them past. If she were distracted by a stressful circumstance, like damage control caused by (say) unbridled truth telling, those words could easily get away from her and she would spout truth after dangerous truth.

"I can't lie!" Remilia said in revelation. "No matter how I try, I can't say statements I know aren't true!" She looked up at Sakuya. "And you have made clearly false statements, and given our long friendship I find it hard to believe the rest of it. Sakuya, we fit this book's description of 'knight' and 'knave.'"

Sakuya had by this time calmed herself. "What is a 'knight' and 'knave', m'lady?" she asked.

"So it seems earnest questions are unaffected..." Remilia observed to herself, then explained what she had learned from the book.

"That is certainly ordinary!" Sakuya said.

Remilia was about to shout that it damned hell was not ordinary before she remembered. "Right. You're a knave... I think."

Sakuya nodded gratefully, visibly relaxing. Then she blinked. "M'lady, there's no letter over there that mysteriously appeared," she said, pointing to the end-table.

Which, being a knave, meant that there _was_ a letter over there that mysteriously appeared. And sure enough, there was.

Remilia opened the letter, as it seemed was addressed to her, but she couldn't be quite sure in this mixed-up world. She thus began reading.

* * *

_Dear Remilia,_

_Two plus two equals four. That at least is true, and the same hand who wrote that sentence is the one that will write this entire message. Don't worry if you don't understand that right now; its significance will become clearer later should you get that far. For the moment, you will just have to trust that this message contains the truth of our situation._

_I trust that you have read the marked place in Smullyan's book, and noted the similarity of your situation that the inhabitants of the Island of Knights and Knaves no doubt find themselves in. You may have already guessed this, but every inhabitant of Gensokyo is now sorted into one of these two groups, __**knight**_ _and __**knave**__, and are referred to in the same vocabulary._

_Knights, of course, always utter what they know and believe is the truth and knaves can only utter contrary to their knowledge and convictions. This caveat will become important later, should you reach it, but for now we can crudely say that every statement that a knight says is the truth, while any statement a knave says is a lie. If the individual's knowledge does not allow them to know the truth or falsehood of their statement, then they cannot make that statement. If the individual's knowledge does not allow them to know the correct answer to a question, they cannot answer and risk answering contrary to their nature. It is now the nature of anyone who is either a knight or knave, and from now on any person you meet is either a knight or a knave, referred to as their __**type**__._

_You cannot, under any circumstances, force a knight to state a lie, or a knave to utter the truth. Nor can you force a knight into action to make lie of his or her words, nor force a knave into action to make his or her words the truth. A knight or knave may ignore at his/her discretion any question you already know the correct answer to, but in general a denizen of Gensokyo will answer any other question. (That means, for instance, you cannot ask someone whether one plus one equals two and expect an answer, since you know the correct answer to that before you asked.) A knight or knave may volunteer such a piece of information —as I have done at the start of this message— but it must be volunteered._

_In general, you are directed to determine the knighthood or knavehood of everyone you meet, when possible. You may ignore the knight- or knavehood of common rabble, such as your own fairy maids. You must, however, prove the knight- or knavehood of all your household (sans fairy maids) to proceed. You may already suspect some of them to be knights or knaves, but you must prove the case either way._

_Good luck._

—_Ran Yakumo_

* * *

"Pfft. An incident, probably caused by the Great Troll of Gensokyo," Remilia mumbled with irritation as she finished the letter. "We'll also probably encounter those other three along the way."

Remilia did find certain appeal in this new kind of incident — one that changed the inhabitants' nature rather than came from an external threat. It was at a change of pace from the usual 'lay the smack down on a random youkai' incident that had become the norm. She might have called herself intrigued had the incident, except for two facts: she was directly affected, and she found logic rather tedious. Like in the Fake Moon incident, the sooner it was solved, the better. She stood in her jammies, with clenched fist, "Come, Sakuya. The game is afoot. You're rehired by the way."

"M'lady, you are wearing your normal clothes."

Which meant Remilia _wasn't _wearing her normal clothes. Indeed, as she looked down, she saw and remembered she was in her jammies. "Let's start after I get dressed and have breakfast," she mumbled, cheeks glowing a bit in embarrassment.

"M'lady."

_(To be continued, only if I'm not a knave.)_


	2. Remilia Questions her Household

_Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.

All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond

Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.

Remilia Scarlet, the Scarlet Devil, has awoken this night to find herself unable to lie, and therefore a knight. So not only is she Queen of the Night, she is also Queen of the Knights. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.

It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.

After breakfast, of course.

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 2:_

_In Which Remilia Scarlet Questions her Household_

* * *

**The Scarlet Devil Mansion**

"What is all this litter?" Remilia Scarlet asked, kicking at scattered beige pamphlets over the otherwise pristine crimson floor. No sooner had she stepped out of her room wearing her pink dress and pouffy hat, she had discovered this mess littering her precious mansion. Her black wings twitched with annoyance at the sight.

"This mess was here when I entered with your breakfast," commented Sakuya, which was, of course, a lie.

"So it had appeared suddenly," Remilia remarked as knelt down to scoop up one of the pamphlets. It was almost ordinary, with two folds dividing it into equal thirds. It had a title proclaiming, "Knights and Knaves: What They Mean To You!" Opening it, Remilia saw that the pamphlet explained, in condensed form, what both the book and the letter explained. Anyone reading the pamphlet would get the basics of the current incident, and a few test questions and statements would reveal what type you were.

"No doubt this is the Gap Hag's doing as well," Remilia (a knight) remarked aloud.

"We will have to explain what 'knight' and 'knave' mean to everyone we meet," stated Sakuya (a knave) falsely.

It still took some effort for Remilia to ignore Sakuya's lies and see the truth behind the lies, but she was getting there. "How convenient!" she said, giving the impression that she never even noticed Sakuya was lying. "And, as it happens, we have found our first group of maids to ask."

_1. The First Dance of Fairies._

The fairies were mumbling over one of the small pamphlets that were now scattered throughout the mansion, and probably the whole of Gensokyo. By the end of the day, everyone no doubt will have a basic grasp of the situation. Probably even the illiterate would get the idea from explanations of their peers.

Remilia crossed her arms and brusquely addressed the three maid-fairies. "You three fairies. I want some information from you." The three cut all their conversation short and obediently stood to attention in line, as was the habit of the mansion.

Remilia stepped up to the first one. She was, of course, one of countless multitudes of fairy maids in the mansion, but even if Remilia did know her personally, there was a new feature to her that she needed to ascertain.

"Are you a knight or a knave?" she asked this first fairy.

The fairy answered, but mumbled indistinctly. Remilia didn't want to get into a one of those horrid cycles of badgering a shy maid to speak up, which would lead her to answer even more quietly. So she turned to the second, asking, "What did she say?"

The second fairy chirped, loud and clear, "She said that she was a knave."

The third shouted, "Don't believe her! She's lying!"

Remilia realized pretty quickly that her question to the first fairy would not have revealed anything probative even if she had heard it, being that knights and knaves alike would claim to be knights. She was, however, able to tell the types of the other two. It was very similar to Smullyan's first knight/knave puzzle, after all.

_Solution: No inhabitant of Gensokyo can now claim to be a knave. For a knight could not falsely claim to be a knave, and a knave could not truthfully claim to be a knave. Therefore, the second fairy is a knave by claiming that the first fairy said such a thing. The third fairy is contradicting her, and therefore is telling the truth and a knight._

_2. What Type is the First Fairy Maid?_

After determining the other two fairies' types (that is, whether they were knights or knaves), Remilia still needed the type of this first fairy. So she addressed the first again. "You, answer this and speak up this time. Are you a lying knight?"

"No," came the first fairy's clear answer.

Remilia was immediately able to tell her type. She also knew such a trick would not work with anyone but her own staff, but Remilia supposed she would pick the low-hanging fruit when she could.

_Solution: There is no such thing as a lying knight, so a knight could not honestly claim to be one. But a knave would, since she isn't any kind of knight, much less a lying one._

_The trick would not work with just anyone, because Remilia knows that there is no such thing as a lying knight. As per (presumedly) Ran's letter stated, someone not of her staff, and therefore had no prior compulsion to obey her orders, need not answer a question Remilia already knows the correct answer to!_

_3. Dismissing the Knave Maid._

Remilia now knew she had two fairy knights and who they were, and as such were reliable sources of information: whatever information they gave would be considered as good as the truth. The fairy maid who was a knave was no longer of any use.

Remilia turned to the fairy maid who was a knave and told her, "You are dismissed."

The fairy hesitated. She waffled around, looking distraught and unable to decide what to do.

Remilia frowned at her, and spoke again, this time sharply. "Didn't you hear me? I said 'you are dismissed!'"

The fairy jumped, but quickly perked up and smiled. She happily moved to clean up the mess that the hallway had become.

Remilia was puzzled for just a moment why the maid was happy and obeyed so promptly the second time, but smacked her head as the answer came to her. Of course she wouldn't know what to do the first time!

_Solution: Remilia had not yet proven she was a knight! 'You are dismissed,' is a statement, and if Remilia were a knave, would imply that the fairy knave was NOT dismissed. But Remilia had said the statement, and the fairy heard perfectly well, so when Remilia stated that she had said the statement, word for word, the fairy knew it was true and therefore her mistress was a knight._

—/—

**The Great Unmoving Library**

"This is going to be tricky..." Remilia mumbled, nipping at her finger.

"I imagine Lady Patchouli would be of no help to us," Sakuya lied.

Remilia was already getting used to the flopped nature of Sakuya's advice. However, the two maids gasped in shock at Sakuya's statement. "The head maid of the mansion is a knave," Remilia found herself volunteering.

The fairy maids perked up immediately at that news, and tittered at the discovery.

Sakuya acquired a dangerous look. Producing one of her knives, she addressed the two maid knights, "You will _not_ stop that, and I will _not_ put one of **these** into an uncomfortable place." The two fairies deciphered the logic quickly and sobered up immediately.

"Enough of this banter! Let's get down to business!" Remilia cut off any further 'discussion' or threats. "Does either of you know if Patchouli is a knight or a knave?"

The two glanced wordlessly at each other, then back at their mistress and boss. "We don't know if Patchouli-sama is a knight or a knave," the first fairy knight said.

The second fairy knight added helpfully, "But we do know that, of Patchouli-sama and Koakuma-san, at least one of them is a knight."

Remilia remembered the advice of the letter, that if the two maids were unsure of the truth of a statement, they could not make the statement. Therefore she could count on at least one of Koakuma and Patchouli being a knight. "Maybe that will be enough. To the library, Sakuya," announced Remilia, dragging Sakuya behind her.

The fairies Sunny Milk and Luna Child sighed in relief as the two left their sight. "Good thing they didn't ask more probative questions, or we might have been in deep trouble. It's hard to pull pranks when you can't lie your way out of tough spots," said the trio's leader. They moved to pick up Star Sapphire. It was obviously time to leave.

_4. Is Patchouli a Knight?_

The doors to Patchouli's great library, Voile, opened with a loud creek. "Sakuya, get someone to see to oiling this door's hinges," she remarked.

"It will not be done, m'lady."

Remilia sighed for the umpteenth time.

"Patchy! Patchy, where are you?" Remilia called out. She immediately thought better of it, "Never mind! Don't answer that! Just call out to us!"

Remilia strained her ears and heard a faint call from the back of the library. It told her everything she needed to know — Patchouli's location. The beauty of it is that she didn't need to know if Patchouli was a knight or knave.

Shortly thereafter, she found Patchouli sitting at a reading table surrounded by books. Koakuma arrived much the same time, carrying a huge stack of books which she just placed on the table. "Ah, there you are, Patchy!" declared Remilia, calling attention to her and Sakuya.

"Remilia," murmured Patchouli.

"So sentence fragments don't seem affected either," Remilia mused. Patchouli moved her head minutely. Remilia couldn't tell, but it looked like a nod. Damn. "You two, make some statements about your type," Remilia said.

Koakuma immediately volunteered. "I am a knight."

Patchouli huffed, removing her glasses. "If that were true, then I am a knave," she stated flatly.

Remilia was able to tell the types of Koakuma and Patchouli with all the information and statements provided thus far.

_Solution: We had been told (by knights) that at least one of Koakuma and Patchouli is a knight. If Koakuma is a knight, then her statement is true. Then if Patchouli's statement were true, it would imply that she were a knave and therefore a knave who has made a true statement. Alternately, if Patchouli's statement were false, then the only way it could be false if she were not a knave, therefore a knight and therefore a knight who has made a false statement. Either leads to a contradiction, and therefore Koakuma is a knave. Since there is at least one of the two is a knight, Patchouli is therefore a knight._

_5. The Maids Were Useless!_

A moment later, Remilia shouted, "ARGH! Those fairy maids' information was _bloody useless!_ I could have deduced your types just fine without it!"

_Solution: An implication is only false if the antecedent (the 'if' part) is true but the consequent (the 'then' part) is false. If Patchouli were a knave, then the consequent is true, and therefore her entire statement is true, and we have a knave who made a true statement. Therefore, it is not possible for Patchouli to be a knave; she must be a knight, and her statement is true. The only way for a true implication can have a false consequent is if the antecedent is also false, and therefore Koakuma was lying and thus a knave._

—/—

**Discussion Within the Library**

"I don't think we can count on _danmaku_ being the solution to this problem," said one Patchouli Knowledge, the Unmoving Great Library of the mansion and a knight. "If the culprit is Yukari as you suppose (and I don't think it's unlikely), it would be a challenge only soluble by using logic, rather than one using bullets. Otherwise, there would be no point sorting us into knights and knaves. Presumably ourselves as knights and knaves can be messed up exactly as easily as our normal selves."

"I agree," Remilia replied. "This isn't to say that _danmaku_ will not be useful, however. One can still imagine troublemakers making things difficult."

Patchouli and Remilia sat across from each other while the two knaves commiserated. Remilia had lost interest in following the conversation, as it consisted completely of lies. It turned out that the books Patchouli had surrounded herself with were ones on logic and logical puzzles instead of her normal tomes on magic.

"Of course." Patchouli abruptly segued from a previous subject: "As you surmised, it seems nothing but a complete statement is affected by type. After all, only _statements_ can be said to be true or false. A command really isn't _true,_ nor is a question, even a rhetorical one. Sentence fragments aren't either, for their meaning is incomplete. Some exclamations can't be said to be true either, though I'm not too sure about them because, given some manipulation, they can be reformed into proper statements, and so fall under the type restrictions."

"So questions and commands can be used without fear?"

"Almost," Patchouli said, leaning back in her chair. "Some commands take the forms of statements, as you discovered from the fairy knave. Further, some questions are really statements in disguise. We'd still best be careful.

"You need to study up on your copy of Smullyan's book, as well," continued the Girl of Knowledge and Shadow, "The puzzles have so far been simple. No doubt they will become more mind-boggling and frustrating for you, as the puzzles in the book had."

Patchouli took a moment to check her notes. She had noted that her handwriting seemed unaffected, but she had been unable to write lies with her right hand. She had assumed this was due to the fact that she was a knight, but Ran's letter made her not quite sure of that anymore. "Now, you said the letter said that you needed to find the types of every household member, neglecting the fairy maids — and of course you're a knight. Don't be so quick in revealing that, by the way. Some of Smullyan's puzzles require you to keep your type a secret, and some challenges may require it as well." The youkai magician held up her hand, fingers but not thumb spread, and ticked them off one by one. "You know the type of yourself, Sakuya, myself, and Koakuma. That's four of the people you need to type. Two remain."

"I'll leave Meiling to last. She's so dumb she'd probably reveal herself as soon as she opens her mouth." Remilia's face was grim at the task looming before her. "That leaves... Flandre. Right now, she's either telling lies to herself, or she's telling herself the truth. Either possibility is frightening."

"She'll probably be in no mood to play logic puzzles, either," observed the witch.

"I might get killed before I even find out if she's a knight or a knave."

The silence passed for a few minutes, as Patchouli thought about the matter. "Perhaps the fairy maids have some information," she said. "They may have heard her mumbling and thus be able to deduce her type." She moved to her great stack of books and started researching, stating in passing, "This may take some time, and you have a book to read."

Remilia collected the small, orange, ragged paperback and stood. "Fair enough. I'll return later."

—/—

**Fairy Twins**

Remilia spent much of the evening sharpening her brain on the strop of Smullyan's book. She wrestled with the various logical concepts Smullyan presented in the guise of logical puzzles, up to and including the subtleties of Gödel's theorems. If she wanted to best Yukari's challenges (if they were Yukari's challenges), she needed to know this stuff cold.

She did not finish the book in one sitting, but Remilia managed to plow through better than a third of it before her mind needed rest. Perhaps now Patchouli had some more information on how to determine Flandre's type.

"Patchy!" she called as she entered the library, "What have you found out?" Hearing no answer, she walked to the reading table where she last saw Patchouli sitting. The magician wasn't visible, but there was a large pile of paper in a disorganized mountain of writing and wood pulp covering Patchouli's chair.

Remilia stared at the mountain of manuscript for a moment. That looked like it could contain...

_"Mukyu!"_ the mountain squeaked, and shuddered. A few leaves of paper shook off and began fluttering down around it.

"Patchy?" Remilia yelped, leaping onto the pile and scattering papers off of it to free her librarian. Slowly, the Locked Girl emerged from the mountain. "Please be careful, Patchy. I don't want you to smother yourself."

"Or concuss myself," added the purple-haired youkai magician, "Darn thing got too tall and fell on me."

"So what did you find out?"

"The first thing I found out that is that Koakuma can write truths —and only truths— if she uses her left hand, while the other can write only lies. Same with Sakuya," Patchouli began.

"Patchy! You're _supposed_ to be finding out stuff about my sister, not wasting time with what a knave can or cannot write!" raged Remilia.

"I'm getting to that. Being able to write truths meant that I could use them to gather statements from our fairy maids," Patchouli explained.

Remilia stared at Patchouli for a moment. "So you could research our maids without stepping foot outside the library, right?"

Patchouli cast her eyes down in shame.

"Whatever. What did you find out, Patchy?"

Patchouli returned better than normal. There was a slight smile on her face. "Through researching more on Smullyan's works and brainstorming my own ideas, then going out and confirming them, I have uncovered an interesting fact about fairies and type."

"Her investigations have revealed that both of a fairy twin pair are knaves, m'lady," Sakuya said, popping out of thin air with a tray of tea and cookies.

"Sakuya, don't muddy the waters!" snapped Patchouli.

"Don't worry, Patchy! I know that means there are at least one of a pair of fairy twins is a knight," Remilia said, brushing off Patchouli's outburst, "being that whatever she says is a lie."

"Incorrect, m'lady," Sakuya replied, before setting down the two cups of tea before Patchouli and Remilia. "These are your favorite blends, but you have each other's favorite."

Remilia glanced at the two cups to confirm her hunch that they were indeed their favorite blends, and they had their own favorites. They did. As expected, while parts of Sakuya's statement may be true, the statement in its entirety was false.

"Sakuya is correct in her understanding, if not her words. At least one of a pair of fairy twins is a knight. I haven't enough data to figure out if other relations are the same way, or different, or if it works for triplets or other multiples, but we at least know that."

"Does this apply to fairies in general, or just to our bunch?" Remilia asked, curious.

Patchouli pushed her glasses onto her nose. "It seems pretty definitive that the fairies believe that it _does_ apply universally to Gensokyo."

"But how could they know such a thing?" asked Remilia in surprise. "Nobody was like this until today!"

"While that's true, remember who we're dealing with here. Smullyan often had puzzles where his special inhabitants knew things that, realistically, there was no expectation for them to know. Yukari might have set things up the same way: the fairies may have knowledge implanted of how family relations within their race behave in terms of type. Or maybe a few of the smarter ones have observed some things about their own race," Patchouli explained the seeming nonsense. "Anyway, the fairy knights are convinced that this is how fairy twins work, and you know what that letter said about statements that someone doesn't know the truth or falsehood of."

"Yeah. If they don't know it, they can't say it," Remilia replied. "So is that it?"

"No. It turns out that only a pair of fairy twins, the Seele twins, knows anything about the type of Flandre... should they exist."

Remilia raised an eyebrow at that last qualifier. "What do you mean by that?"

Patchouli coughed. "What I mean is that, for some reason, the fairies are convinced that one of their number knows _any_thing about Flandre's type if and only if she's one of that unique pair of fairy twins."

"Is this one of those funky 'they just know it' facts?"

"Very likely. Again, remember who we're dealing with."

Remilia raised the other eyebrow. "_Who_ are we dealing with? Yukari... or Smullyan?"

Patchouli had to think about that for a while. "I'm not sure."

"Let's move on. Do you know which pair of fairy twins knows something?"

"No. Like I said, I do not even know such a pair exists yet. It could be that _no_ fairy maid knows anything about Flandre's type."

Remilia was about to shout that there must be someone that knows, given the connotations of the fairies' statements, before she remembered the basic logic governing the incident. Just because you don't have an example doesn't mean you cannot know something about its properties. That was one of the strange things about logic.

"Anyway, I've located all of the fairies that are twins," continued the magician. "We shall proceed to question them on the existence of the Seele twins and their identity. I had a special area for this prepared in advance." With this Pachouli rose and bade Remilia to as well.

"Very well," Remilia said, and rose to follow.

_6. Vermillion Bloom and Dream Flower_

"Patchouli..." the vampire mistress of the mansion, "...the twins... each pair looks alike!"

"Of course they do. They're twins," came the purple bean sprout's nonchalant answer.

"Fairy twins are identical?"

"No other kind." Patchouli looked up at the two fairy twins, with identical features that could not be distinguished from each other. "First on the docket: Vermillion Bloom and Dream Flower. Establish your identities, if you will," Patchouli began. Like Remilia, she knew that there was no expectation for their answers to be useful.

"I'm Vermillion Bloom," the first twin said.

The second twin picked up. "If that's true, then I'm Dream Flower!"

The first twin returned, "If that's true, then you're a _bitch!"_

The second twin growled heatedly, "There's only one bitch between us both, sister!"

"Please, ladies," Patchouli cut off any potential sibling rivalry that had been brewing, "all we want to know is what you know about the Seele twins!"

The first twin huffed. "We know that the Seele twins exist!"

"No, we don't!" the second countered.

Patchouli sighed, slumping against her bench. These two were of no help.

_Solution: The two twins contradict each other on whether they know whether the Seele twins exist, and are therefore of opposite type. If the first twin is Vermillion Bloom, then indeed the second twin is Dream Flower. Therefore, the second twin told the truth and is a knight, this makes the first twin a knave and therefore the twins do not know that the Seele twins exist._

_7. Crimson Gale and Holy Wind_

"Next is Crimson Gale and Holy Wind. Please establish your identities."

The two twins stood at attention to Patchouli's instructions.

"I'm Crimson Gale," said the first.

"If that's true, then I'm Holy Wind," said the second.

Patchouli's hand slammed against the wooden bench. "We're _not_ going through **that** game again! All we want to know is if the Seele twins exist!"

The second answered, "They exist if and only if Diamond Dust and Star Ruby are both knights."

Patchouli perked up.

"What's the matter?" asked Remilia.

"Those are the names of one of the remaining twins we will interview! This is a big break!" said Patchouli in genuine, if somewhat subdued, excitement.

_Solution: Like the preceding problem, if the first twin is Crimson Gale, then the second twin is certainly Holy Wind, and therefore the second twin's statement is true and she is a knight. We do not know the identity of the twins, but we do know, from the second twin's second statement, that if the twin fairies Diamond Dust and Star Ruby are both knights that the Seele twins really exist._

_8. Diamond Dust and Star Ruby_

Patchouli took out a new sheet of paper. "Bring in Diamond Dust and Star Ruby," she ordered. Two identical fairies were ushered in once more. "Please establish your identities," Patchouli ordered.

The first twin stated, "I am Diamond Dust."

The second twin stated, "I'm Star Ruby."

Patchouli blinked, then asked, "Do you know who the Seele twins are?"

The first twin answered, "Yes."

Patchouli smiled. She was a proverbial stone's throw away from her answer.

_Solution: If the first twin is really Star Ruby, then the second twin is Diamond Dust. But that would require both of them to lie, and therefore both be knaves. But we know that, as twins, they are __**not**_ _both knaves. That means the first twin really is Diamond Dust, and the second twin really is Star Ruby. They are both knights. As established in the preceding problem, if both are knights then the Seele sisters exist. Patchouli realized this, and so her question was fair, and the twins do indeed know who the Seele twins were._

—/—

**Finally, The Seele Twins!**

Remilia stood abruptly, disrupting the proceedings. "You know who the Seele twins are?" she shouted, "WHO?"

"We are they," Diamond Dust said.

"TELL ME WHAT YOU KNOW ABOUT FLANDRE!" screamed the vampire, her eyes glowing with unearthly energy.

Diamond Dust was not easily cowed. "Keep your blouse on, mistress. We will tell you just after and only just after you identify Crimson Gale's type," she said calmly.

"Why you..."

"Remilia, please _sit_ down, and _calm_ down," came Patchouli's sharp, commanding voice. "They'll do no good for us if they're dead." She turned to Sakuya. "Sakuya, bring back both Crimson Gale and Holy Wind."

"I will not," said the Knavish Maid with a bow.

_9. Was Crimson Gale Lying?_

As the twins were brought in, it occurred to Patchouli and Remilia that discovering the type of Crimson Gale would not be so simple. "I don't know who the knight is!" Remilia whined.

"There's no guarantee that we'll be able to re-identify either," Patchouli murmured miserably. "I didn't imagine needing them again, so I didn't bother to keep track of their type. A mistake. My apologies, Remilia-sama."

Remilia slumped in her chair. "And since all we figured out was that one was a knight, we're right back at the start."

Patchouli let out a cleansing sigh. It was time to start from the beginning and hope that she would be able to figure out who was the knight and knave and who was Crimson Gale.

Before the witch could begin, however, Remilia sat up erect in her chair, shouting in sudden insight. "WAIT! I know Crimson Gale's type, and I don't even need any more testimony!"

_Solution: In problem 7, if the first twin really was Crimson Gale, then of course she was telling the truth. If the first twin was not telling the truth, then she was not Crimson Gale; her sister would be Crimson Gale, who was, of course, telling the truth. In either case, Crimson Gale never lied and is therefore a knight._

–/–

**Flandre was Her?**

"Excellent deduction, mistress!" said Diamond Dust cheerfully. "As promised, we shall reveal what we know about Flandre."

Remilia was too happy with her sudden, brilliant success to be impatient about getting the information. Patchouli was secretly proud of Remilia. Although the magician would probably have hit upon the idea herself, Remilia's sudden insight allowed her to beat her to the punch. That was exactly the kind of insight the mistress would need to solve this incident.

From here, Star Ruby picked up. "Sadly, it is not very much information. Your sister has made her most probative statements while she had cast Taboo 'Four of a Kind'. We were only able to figure out one thing about her type: in the spell, all the copies are of the same type — either all knights or all knaves. We do not know the type of Flandre herself, or if it must be different from that of her clones."

Remilia collapsed onto the desk, completely spent. It was such a small piece of information, how in the world could she expect to meet Yukari's (?) challenge now?

Obviously, only one way. She had to face Flandre...

—/—

_10. Four of a Kind_

Remilia stood in front of the vault in the basement. Beyond lay a creature of frightful power and terrifying insanity. Beyond lay the secret of the Scarlet Devil Mansion. Beyond the vault was one Flandre Scarlet, the Diabolic Wave.

Remilia's dear, insane younger sister.

The vampire mistress of the mansion wondered how many damned lies, or unsettling truths, had passed Flandre's lips since the incident began, with only herself to listen to them. Did she know they were only lies, or only truths? Which would be worse? She still had no answer.

"Come in, dear sister!" came a quartet of voices, singing ever so slightly off-key.

Remilia gulped. She quickly dogged the vault door open and used her vampire strength to easily pull it open. She stepped in.

The room was as bad as it ever been. The floor was a spiderweb of cracks and strewn with rubble. She didn't see any of Flandre's toys. Probably smashed to dust. Furniture was in disrepair, as if meeting a wrecking ball.

A familiar beige pamphlet fluttered down before her. She snatched it out the air, reading the familiar title, "Knights and Knaves: What They Mean To You!"

The Gap Hag had loused things up yet again!

Flandre knew she now only spoke truth or lies. There was no way to pretend that the comfortable lie was a hopeful truth, or to pretend that the hurtful truth was just a pessimistic lie.

Remilia dreaded looking up, already knowing what she would find. Four copies of her sister dressed in blood-red. Her crimson eyes burned with a seething madness that had been flamed by destructive feedback of her lies, or her truths.

"Shall we play a game? A _logical_ game?" cooed the blonde-haired head full of crazy.

Wordlessly, Remilia took to the air, the only way she could meet Flandre's altitude.

"Can you tell who is the real me? Can you tell if that real me is a knight or a knave?" tittered the quadruplet of Flandres sinisterly as they fluttered about the room.

"I will certainly try, Flandre! I will try to save you! Issue your challenge!" barked the elder sister.

The clones erupted in a cacophony of laughter as they let loose a dense hail of _danmaku_ bullets. Remilia got to dodging the deadly missiles, focusing on the movement of the deadly projectiles.

"I wonder if you're a knight, and your words the truth, sister," chuckled Flandre's voice as Remilia dodged. She couldn't keep track of the speaker at that moment. She supposed that it wasn't really important. "Very well! Look up now!" came her abrupt command as the fusillade let up quite a bit.

Remilia did as she asked. The four were lined up above her.

Flandre 1: "There is exactly one knave amongst us!"  
Flandre 2: "There are exactly two knaves amongst us!"  
Flandre 3: "There are exactly three knaves amongst us!"  
Flandre 4: "All four of us are knaves!"

Remilia was grateful that each clone had a numbered badge pinned to her blood-red vest, labeled one through four. Flandre had enough presence of mind to not make things more difficult than they already were.

"Grab the one you think is the real me, and declare me a knight or knave!" they cackled together. Their laughter ripped the air again, while _danmaku_ cut through it.

Both Remilia's logic and her _danmaku_ grazing were slowed and dulled, as she was doing both at once. She had retreated to a spot where her spell card was tamer and did not require as much concentration to effectively dodge, allowing her to spend more brainpower on the problem Flandre had posed.

_Think, Remilia! You know there's a way to solve it!_ thought the Eternally Young Scarlet Moon.

"Hurry, sister! You have until the end of the spell card to figure it out!" cackled Flandre's voice. Remilia didn't catch which clone said it, but she hadn't figured out what type any of them were. Remilia figured she'd best assume that she had to hurry it up.

It was a long moment before Remilia finally cracked both Flandre's puzzle and her _danmaku_. The elder vampire became a missile and screamed through the curtain of death, artfully dodging the projectiles. As she neared the clone she thought was the real Flandre, the bullets got denser and scraped her face and body.

Remilia clapped her arms around the chosen copy — clamped tightly. The elder sister almost tore her jaw apart as she screamed into this Flandre's ear, _"YOU. ARE. A..."_

_Solution: Obviously, at most one Flandre can be correct in the number of knaves, so there are at least three knaves amongst them. But there cannot be four knaves, because otherwise Flandre 4's statement would result in a true statement from a knave. Therefore, Flandre 4 is a knave and there is exactly three knaves amongst them. Consequently, Flandre 3 is the unique knight amongst them because she states the correct number of knaves, and since all the FoaK copies must be of the same type, Flandre 3 is the real Flandre, and she is a knight._

—/—

_"YOU. ARE. A. KNIGHT!"_

_**~Spell Break~**_

The spell disappeared around them as Remilia held the real Flandre in her arms and announced to the heavens her type. She had chosen correctly.

"Sister!" Flandre wailed, and returned the hug as they spiraled down to the earth. "I was worried I would kill you!" she sobbed.

"I was scared for my life as well. However, you didn't kill me, Flandre," Remilia whispered back, "I love you, Flandre."

Flandre was confused, and then made the proper deduction. Her sister was a knight, too. Her words were the truth.

Remilia was glad to see Flandre like this. The madness in her eyes burned only slightly now, relieved by the great truth of her life that quashed all of the little, hurtful truths. Her sister had heard her, and rescued her. And loved her.

Remilia held onto Flandre until the girl sobbed herself happily to sleep. The elder sister then made a makeshift bed out of the remains of the old and some blankets. With her sister wrapped and snug, Remilia moved towards the open vault door.

"G'nigh', sis. L've you," murmured Flandre.

"Good night, Flandre."

—/—

**Interlude: Is Vermillion Bloom a Bitch, or is Dream Flower?**

"You still up, Patchy?" asked Remilia.

Patchouli clapped a manilla folder closed, containing proceedings from the Fairy Twins interviews. "Just satisfying some curiosity. It was a little something that was bugging me about the interview of our fairy twins. Specifically about Vermillion Bloom and Dream Flower."

"And that was?"

"Who, if either, was the bitch."

Remilia's eyes crossed as she made a confused squeak. "Is it important?" she asked.

"No. As I said, it was bugging me. Let me show you." Patchouli pulled out a white piece of paper with a neatly typed transcript:

_1. Twin 1 / I'm Vermillion Bloom.  
2. Twin 2 / If that's true, then I'm Dream Flower!  
3. Twin 1 / If that's true, then you're a bitch!  
4. Twin 2 / There's only one bitch between us both, sister!  
5. Patchouli / Please, ladies, all we want to know is what you know about the Seele twins!  
6. Twin 1 / We know that the Seele twins exist!  
7. Twin 2 / No, we don't!_

Patchouli pointed toward that last line. "At this point, I deduced that the twins had nothing useful to add and dismissed them. I didn't think anything of it until after you confronted Flandre, and I wanted to find out which one was which, and more importantly, who was the bitch."

"The two twins contradict each other in the last two lines, so they can't both be knights," Remilia observed.

Patchouli nodded. "And the first two lines give us that the second twin is a knight, since if the first twin really was Vermillion Bloom, then the second twin would indeed be Dream Flower. Together with your observation, we conclude also that the first twin is the knave."

Remilia grinned, fangs gleaming. "And line four, said by the knight, means that there really was only one bitch between them."

"Since line three was said by the knave, we know that her statement was false, and therefore the consequent was false, so the knight was definitely not the bitch... the knave was."

"Finally, since the knave claimed to be Vermillion Bloom, it is false. She must be Dream Flower. So Dream Flower is a lying bitch, while Vermillion Bloom isn't," finished Remilia.

The two looked at the transcript for several seconds. "You thinking what I'm thinking?" asked Patchouli.

"If you're thinking of reasserting of our domination over the maid staff, then you're thinking what I'm thinking," Remilia stated truthfully with a cruel smirk. This should be of no surprise, since she was, after all, a knight.

"I'll get the blank banners."

"I'll get the paint."

_(To be continued... unless I have exactly $11 in my pocket.)_


	3. Remilia Scales Youkai Mountain

_16Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

The red moon beamed down on the Scarlet Devil Mansion as Remilia Scarlet, the Eternally Young Scarlet Moon, perched atop the spire that was the highest point for tens of kilometers about her. The red mist she had spread throughout the land would enable her to walk out in sunlight. However, the lack of said sunlight seemed to trouble humans, which is why two of their number were now at the mansion, on the flat roof facing her.

There was the local miko, a young thing of fourteen tender years and dressed in white and red livery, one Reimu Hakurei, the Wonderful Shrine Maiden of Paradise. There was a mischievous human magician with her, dressed in a black-white witch costume with a black wide-brimmed pointed cap and riding a broom, one Marisa Kirisame, the Ordinary Magician.

Surveying these strange intruders, Remilia purred, "Looks like it's going to be a fun fight. And it looks like it's going to be a long night."

Just as the mistress of the mansion was about to begin the _danmaku_ battle that would decide the fate of the sun over Gensokyo — whether it would emerge once more, or remain forever hidden behind the veil — Reimu held up a hand. "Wait! Before we begin, there's something I wanted to know."

Remilia paused to consider, but then nodded in agreement. "Very well. Ask."

"What is the origin of the pink color of your dress?" asked Reimu out of sheer curiosity.

Remilia was about to announce that the pink color was the faded stains of the blood of her many victims, being a vampire and all. But instead of this shocking revelation, a much more mundane and truthful one emerged.

"It's just a pink color, because I'm a little girl!"

Remilia's hands went to clamp over her mouth as her eyes grew wide. The secret was out — the coloring was nothing but ordinary dye, suitable for a little girl like her (even if she was 500 years old). Any bloodstained clothes she owned have long since been replaced, and Sakuya kept all her clothes scrupulously clean.

The two humans were trying desperately to hold in their laughter. Their cheeks puffed and their eyes were wide and tearing up at the pain of withstanding their spasms. Suddenly, neither could stand it any longer, and burst into peals of hilarity.

"Marisa! Marisa! Did you hear that! She's a pink clad little girl!" gasped Reimu through gut busting laughter.

The magician overbalanced at that point, depositing Marisa onto the ground. She pounded the concrete with her fist, burying her face in the crook of her other arm. "I'm dying! I'm dying of laughter!" she gasped, sounding like a suffocating hyena.

"S-Stop it! Stop laughing!" shouted the vampire kid, desperately trying to regain her charisma. Her eyes burned, and tears spilled from them. _"Take me seriously, damn it!"_ she screeched. However, the two only intensified their laughter, which Remilia didn't even think was possible.

It all proved too much, and Remilia squatted down gripping her mob cap and let out a pitiful keening wail, _"U~~! U~~!"_

_**~THUD~**_

Remilia blinked her eyes open. She was in her usual jammies, partially sprawled across the floor right next to her bed.

"We _gotta_ get this thing resolved!" she mumbled, as the Scarlet Devil resigned to the fact she wasn't going to be able to sleep through the day.

* * *

In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.

All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond Smullyan.

Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.

Remilia Scarlet, the Scarlet Devil, has awoken this night to find herself unable to lie, and therefore a knight. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.

It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.

This day, Remilia has woken prematurely from her nightmares (knightmares?) of people in Gensokyo learning the horrible secret of the Scarlet Devil, and consequently be called the 'Frilly Pink Vampire.'

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 3:_

_In Which Remilia Scarlet Scales Youkai Mountain_

* * *

"Morning, m'lady," one Sakuya Izayoi, the Perfect, Elegant and Knavish Servant, said in greeting, setting Remilia's breakfast in front of her mistress. It was an ordinary breakfast, consisting of a light meal with more grains than meat, perfect for a stomach only just gearing up for the day.

Of course, Sakuya, being a knave, would maybe call it midmorning tea.

Or dinner.

Or boiled ox-hoof.

"This is your tartan electric massage unit with sunlamp attached."

Or that.

"Taking a nosedive right into the ridiculous, Sakuya?" came Remilia's rhetorical question.

"No, I'm not. This is a perfectly reasonable description of your breakfast," Sakuya replied with a wide grin on her face, "It's not amusing at all, either."

"You are having _way_ too much fun with this," Remilia observed.

"Better to have way too much fun than not enough!"

Remilia looked up to find one Flandre Scarlet, Sister of the Devil (that is, of Remilia herself), sitting at the other end of the table with a cute little lobster bib tucked into her collar. Flandre had driven herself mad in the basement of the Scarlet Devil Mansion, confined there due to her awesome destructive powers. But currently, she seemed as sane as anyone, her cute girlish face beaming as she held fork in one hand, spoon in the other. It was like she was about to childishly bang the table, demanding her food under threat of holding her breath.

"Flandre..." Remilia began carefully, "...why are you outside the basement?"

"Sakuya told me that you'd be super-glad if I joined you for breakfast!" came her chirping reply.

"Flandre..." Remilia began even more carefully, "...have you fully realized that Sakuya is a knave?"

Flandre's face fell just a fraction. "...You're not super-happy that I joined you for breakfast?"

"Not _super_-happy. More apprehensive," Remilia corrected, trying to navigate around her knight nature, to take the edge off a normally sharp truth. "You have tremendous power to destroy things. Your past antics are proof enough of that. If they were more under control, I would be more amenable to letting you out of the basement."

Flandre looked downcast while Remilia broke down the problem and a few moments after, but then she perked up again. "How about I work on my control!" she shouted triumphantly.

Remilia paused in eating her breakfast. "That _sounds_ like a good idea, but I have no idea how one goes about taming the power of destruction," Remilia admitted.

"I'll just go ask Patchy!" Flandre announced. Then, she asked with concern, "Is Patchy a knight or a knave?"

"Knight. But Koa's a knave, so be careful." She was not about to let Flandre play any more logic games (where she could get frustrated) until she learned more control.

Few words were spoken as they finished their meal. It was hard to catch up on four hundred years of being locked away in one sitting, but baby steps would carry them for now.

Remilia quietly nursed her tea after finishing breakfast, enjoying the quiet moment after a meal before starting the day's activities. Remilia was almost startled out of her skin as a pair of deceptively frail arms hugged her neck.

"I love you, sis. And because I'm a knight, that's the truth!" giggled Flandre, "And because _you're_ a knight, I know you love me!"

Remilia let out a sigh of exasperation. "Hey. Don't be spreading it around, okay?" replied Remilia.

"'Kay!" the younger vampire chirped as she pulled away, then pranced away, off to the library.

_She'll probably scare what little blood Patchy has out of her. Maybe this knight/knave thing isn't so bad after all,_ thought the mistress of the mansion, as she took a further sip of tea.

The sudden image of Marisa and Reimu laughing their guts out caused her to spew that tea all over the table. _I take that back!_

"Oh my! Spewing tea befits a lady of _your_ level of elegance, m'lady!" came the maid's slightly teasing voice as she set about wiping up the mess.

Remilia _knew_ Sakuya's words were a lie. She _knew_ that they weren't her faithful maid's true thoughts, and that she was chastising her for such unladylike behavior. But she'd be damned if she for one minute though that Sakuya wasn't milking her knavehood for all the amusement it was worth.

—/—

"Patchy! Where are you, you recluse?" shouted Remilia as she entered the great library again.

"Here!" came a call from the opposite end.

Remilia quickly located Patchouli. The bookworm was surrounded with small jotted notes. "That was a dirty trick, mistress," came a flat voice that wasn't even a growl, "The younger sister gave me a fright."

"It is the mistress's prerogative to play tricks on her household," reminded Remilia. "Were you able to help her?"

"I have some ideas for Flandre's training," admitted the purple-haired bean sprout, "I don't know if _any_ method will help. We can only try." Patchouli sat up, a more animated look on her face. "For now, I have something intriguing to show you."

"What is it?"

"Observe, mistress." Remilia watched as Patchouli pulled out a small card and began writing on it with her left hand, somewhat awkwardly curling it around so that she did not smudge the writing. "Read this, please," Patchouli requested, handing Remilia the card. Remilia did so.

_Remilia Scarlet is human._

Remilia blinked, then her eyes grew wide. "Patchy! This sentence is _false!_"

The magician nodded. "Indeed."

"But you're a knight! You _must_ utter the truth!" Remilia protested, feeling as if the one thing she could be sure of shift beneath her feet.

"But the written word is _not_ an utterance!" Patchouli said in correction of Remilia's obviously wrong assumption. "That's one of the facts I have discovered about how writing works in Gensokyo in this incident. So far, everyone whom I've tested (and who are literate), have one hand that can only write truths, and another that can only write lies. That applies to knights and knaves alike.

"Unfortunately, _which_ hand writes the truth and which hand writes only lies is random for each person. For me, the truth-writer happens to be my right hand. For Sakuya and Koakuma, they are their left hands."

"Sakuya was right handed before yesterday," Remilia interjected.

"As was Koakuma, but the former dominant hand now has nothing to do with which hand will now write lies and which writes the truth. I've tested all our literate staff, and there is an even split between those whose right hand is knighted and whose right hand is knaved, and they bear no clear relation to the former dominant hand.

"Everyone is now ambidextrous in terms of how well they write. What's more, it seems to be _complete_ ambidexterity! I can't tell the difference between sentences written by my left hand and with my right, other than by content. Same for anyone literate in this household — except maybe you, since I don't even know your present handedness. I've taken to calling a literate who is a completely ambidextrous a 'symmetric literate,' just to avoid the clunkiness of 'completely ambidextrous literate being.'"

Remilia frowned, suddenly sour as a realization came to her. "There's going to be a puzzle based on this, isn't there?"

"I wouldn't bet against it, mistress," Patchouli advised. A moment later, she asked, "Have you gotten another message relating to our next challenge?"

"No."

Patchouli quirked an eyebrow in curiosity. "Presumably Yukari, or whoever is orchestrating this incident, is keeping tabs on your progress. If you had completed your task, we probably should have gotten another message by now."

"I think I did complete my task. I found out the types of myself, you, Koakuma, Sakuya, and Flandre. All the rest are fairy rabble."

"You realize someone's missing from that list."

Remilia was silent for several seconds. "Ah. China."

—/—

**The Chinese Gatekeeper**

_11. The Safe Key_

"Do we even have a room in the mansion for her?" asked Remilia in a sort of general sense. They were outside the Scarlet Devil Mansion building, walking along the path toward the main gate in the outer wall. The sun was high in the sky, such that it would have been the teensiest bit deadly to Remilia to be standing in it without shade. Fortunately, shade she did have in the form of a parasol.

"She is probably dutifully standing at attention at the front gate," sighed Sakuya as she lied at Remilia's side. Remilia's translations were getting to be automatic: _That means the layabout's probably asleep at her post._

The two stopped as the gate and its surroundings became clear. The tall form of the Chinese Girl stood erect in front of the main gate. As the wind wafted her blood-red hair behind her, she glared intensely at the two.

Remilia stared at the sight for a moment, then said, "Sakuya, I think Meiling has made a truth-teller out of you!"

"Buh? Wha? _Bfft,_" sputtered the lying maid.

Remilia stepped forward to address the gate guard. "Meiling, make a statement to prove yourself a knave or a knight," she curtly ordered.

Meiling started shaking as she pressed her lips firmly together and said nothing.

Remilia frowned. She couldn't tell Meiling's type if she didn't speak. "Answer me, you easy-going gate guard!" she barked, noting the softened insult. She had meant to use 'useless' instead of 'easy-going.' It seemed that her insults were tempered a bit by her knighthood. She had a hunch that knaves weren't similarly affected (unless the target truly deserved the insult).

Nevertheless, the attentive and concerned easy-going gate guard remained silent, pressing her lips so hard together that they were turning white. She then pointed toward a sign that had been set up not far from two marble pedestals.

* * *

_To Whom It May Concern,_

_I yet know of no logical way to completely prove the truth of this sign. You will just have to trust me that this sign is the complete truth._

_Before you stands your gatekeeper, Hong Meiling, and two pedestals, each with a key laying on them: gold and silver. The gate to the Scarlet Devil Mansion is locked and a robust shield surrounding the property that prevents both entry and exit. One of these keys will open the gate (at the same time lowering that robust shield), while the other will result in locking your type forever. That is, if you are a knight, you will remain a knight even after the incident concludes; similarly if you are a knave._

_Your gatekeeper has been instructed, under pain of infinite everlasting pain, not to reveal which key does which, or interfere with their use until the test is over. She is only allowed, again under pain of infinite everlasting pain, to answer one yes-no question about the keys and otherwise be silent until the test is over. She is, of course, a knight or a knave._

_Good Luck._

—_Ran Yakumo_

* * *

"Oh, for..." Remilia spat. Not only did she have to determine the type of all the household, but she also had to use logic to leave her own estate?

"Allow me to be the test of the keys, m'lady! I would mind very much to remain a knave for eternity in your place," offered Sakuya.

Remilia held up a hand, stopping Sakuya from taking more than a step. "That will not be necessary. We don't need to know Meiling's type to solve this."

_Solution: Remilia asked, "Were I to ask you, 'Is the gold key the one that opens the gates?' what would be your answer?"_

_Without loss of generality, suppose Meiling's answer to the straight question 'Is the gold key the one that opens the gates?' is 'yes.' If Meiling is a knight, this is the truth — the true answer is 'yes;' if she's a knave, it is a lie — the true answer is 'no.' But Remilia did not ask this question, she asked what Meiling's answer to the straight question would have been had she asked it. If Meiling is a knight, then the true answer to the straight question is 'yes,' so Meiling tells the truth and answers 'yes.' If Meiling is a knave, then the true answer to the straight question is 'no,' so the false answer —the one Meiling would have given if asked the straight question— is 'yes,' so she would lie and answer 'no' to Remilia's question. In both cases, Meiling's answer would be the true answer to the straight question. The same holds for if Meiling's answer to the straight question is 'no.'_

—/—

Remilia took the silver key and confidently inserted it into the gate lock. Sakuya and Meiling held their breaths, even though Meiling knew the correct key. Remilia turned the key, and the lock made a satisfying click as the shield around the estate dropped immediately. The silver key was indeed the correct key.

While Sakuya elegantly released the breath she'd been holding, Meiling let out a loud roaring gasp of air being drawn into her volumous lungs. "Hyah! I'm still alive!" she bellowed.

"Ah. She's a knight," announced the vampire knight, noting the obviously true statement. She wondered if her supposition yesterday that Meiling would reveal herself with her first sentence counted, as she actually managed to keep her type secret for one answer to a question, albeit not of her own volition.

Sakuya got this wicked grin on her face. "No, Meiling. You're not alive anymore. This is hell, and I'll be your nightmare for the rest of eternity."

_Oh, that's cruel! Bravo, Sakuya!_ "As I said before, _way_ to much fun," Remilia announced, although her face bore an evil smile.

Meiling let out a frightened screech as Sakuya's lies hit home. _"KYEEEE! SAKUYA-SAN!_ Or is it a demon from hell that just looks like you? I mean, like the head maid, Sakuya? Actually, I think I may prefer a demon to Sakuya-san!" she babbled.

"Meiling, shut up." Meiling's mouth closed with a sharp click. "Sakuya's just teasing you because she's a knave."

"'Knave'? What's a knave? What's this all about?" asked Meiling, disoriented by the strange words.

Remilia cocked her head to the side. "Meiling, did you read a beige pamphlet about knights and knaves?"

Meiling vigorously shook her head. It rattled.

_"Were_ there any pamphlets?" Remilia pressed.

"Sure. All over the place. I spent most of the night gathering them up and burning them."

"All those pamphlets, and you didn't read a-one..." sighed the mistress. "I'll see to it that you get a copy. You will, however, read it, or I'll have to punish you personally." A fanged grin spread across her face. "I'm half hoping you'll disobey me, Hong Meiling; it's been a long time since I've _properly_ punished one of my servants."

Meiling made an inarticulate strangling sound of terror.

Remilia wheeled about and started back towards the mansion. "Come, Sakuya. I think the next message from the fox will be waiting for us now."

—/—

**Ran's Second Message**

Sure enough, lying on Remilia's end-table there was a sealed envelope addressed to her that no fairy maid could honestly recall being there just minutes before.

* * *

_Dear Remilia,_

_I apologize for being unable to verify the truth of any of these documents, calling upon you to place faith in them that they are true. Yukari-sama left many of the details of this incident to me, and have not yet come up with a logical means of proving these messages' truth. As of now, you only have my solemn promise that I have written all of these messages with the same hand (my 'knighted hand') in accordance to the recent discovery by your staff magician, Patchouli Knowledge._

_Now that you have identified all of your household (sans fairy maids) as knights or knaves, you may now turn your attentions outside the Scarlet Devil Mansion. This will require you or one of your staff to leave the mansion to carry out these missions. Furthermore, you personally are likely the only one who can reliably do so. For Sakuya and Koakuma will have difficulty answering challenges due to their knave nature, Patchouli and Flandre are as yet too unhealthy and too unstable respectively to be trusted with missions involving long journeys, and Hong Meiling will probably only barely grasp that —for the time being— she can only utter truth, much less solve puzzles._

_And before you get ideas, Yukari confiscated the magics that allowed Patchouli to communicate with Marisa during the Subterranean Sun incident. I apologize for my master's crass theft._

_You are to go to the Moriya Shrine at Youkai mountain. As always, you must determine the type of all the inhabitants of Youkai Mountain, sans rabble (consisting of fairies, kappa and tengu). In addition, you must settle the question of whether or not Sanae is a good girl, and whether or not Aya Shameimaru will write the truth in her next Bunbunmaru edition. In addition, the kappa are up to something, and it would behoove us all to find out what they're doing._

_Yukari-sama has just reminded me none of your retinue have visited Youkai Mountain. You might want to ask the miko or the black-white about the people there._

_Good Luck._

—_Ran Yakumo_

* * *

_"MUKYUUU!"_ came a cry of rage from the library.

Remilia didn't like the sound of that, so she literally flew to the library, bursting through the door. "Patchy! What's wrong, Patchy?"

The library was in disarray in this section. Books on magic were strewn willy-nilly. Notes on magic were scattered almost everywhere else. Patchouli was almost burrowing through a particularly unruly pile, shouting as she threw book after book over her shoulder, "I haven't found it! Despite my searching, I haven't found the magics that allowed me to communicate with Marisa, and I'm running out of places to look!"

"The Yakumos have it. I just read the message. It isn't here."

Patchouli's face grew wild-eyed with anger. Remilia felt a drop of sweat roll down her forehead; she'd only seen a look _that_ mad on Flandre — granted, it was one of her younger sister's saner moments, but even that was quite mad. "If I could make some suitably nasty revenge stick, I would," she growled darkly, just before going into a horrid-sounding coughing fit.

Remilia smiled faintly. For a knight, it was as good as a promise to send someone to Shikieiki's court.

Patchouli finally got whatever was in her throat out, and spoke in a hoarse whisper, "That means you're on your own for solving the puzzles that will no doubt encounter in this incident."

"Ran says much the same thing," Remilia confirmed Patchouli's suspicion. "Don't worry. I may not be as smart as you, but I'm no dummy."

"I hope you are up to the challenge," came the witch's reply. "If you are to go to Youkai Mountain, you best know something about the inhabitants. A visit to the Hakurei Shrine or to the black-white's house would be most appropriate."

"Do you seriously believe Reimu or Marisa will sit still while this incident is going on?" asked Remilia rhetorically.

The reclusive witch made a thoughtful noise before answering. "Point taken. I'll compile what my research turns up into a brief for you to read," Patchouli said. She pointed sharply to Remilia, ordering as firmly as her weak lungs allowed, "And get some rest. I _know_ you didn't sleep well."

"I will..." Remilia murmured as she shuffled out the library.

—/—

**A Snake, a Frog, and a Girl**

A few hours' siesta and a lunch later (Sakuya called it a 'metaphysical treatise on rotary engines'), Remilia found that Sakuya had prepared for the day trip to the Moriya Shrine. Koakuma jogged up to the two, and handed them a portfolio she referred to 'Mistress Patchouli's yaoi doujin.' Remilia grew suspicious: why _that_ particular lie?

Remilia put that out of her mind as she and her maid took to the air, the vampire careful to always keep her parasol such that she was in its shade, while the human carried the bulk of the supplies. Fortunately, high noon meant that at no point did she need to obstruct her vision too badly.

The flight to the Moriya shrine was uneventful, if further than the mansion to the Hakurei shrine. This surprised Remilia because Patchy's report indicated she should encounter several gods and the kappa along the way.

It seemed they were expected, as one Sanae Kochiya, Skin-deep and Shallow Human, looked up and waved them down. She said not a word as she lead them to the shrine building, where one Kanako Yasaka, the Avatar of Mountains and Lakes, and Suwako Moriya, the Epitome of Native Gods, sat on the veranda. Sanae gave them a sharp, warning look before taking a seat by Kanako, opposite Suwako.

Kanako rolled her eyes at Sanae's warning look and asked, "What brings you here in the middle of the day, vampire?"

Remilia felt the buzz of still-strong deities gently urging her her to leave, even though she had been invited. "I came to ask you a question, and get your types." The three exchanged puzzled, unsure glances. "Ugh. One plus two equals three," she said in resignation.

Remilia's knighthood proven, Kanako said, "Ask your question, then."

_12. Is Sanae a Good Girl?_

"Very well. Is Sanae Kochiya a good girl or not?" Remilia asked.

Sanae immediately proclaimed, "I am a good girl!" She the turned to glare at the two goddesses.

Kanako and Suwako glanced at each other. Then Kanako said to Remilia, "Among us three, at least one of us is a knight, and at least one of us is a knave."

"That is true!" Suwako agreed, nodding her head.

Remilia grinned toothily. She might not know everything about the Moriya shrine, but she had her principle answer.

_Solution: Kanako and Suwako agree, so they are either both knights or both knaves. If Kanako is a knight, then her statement is true and therefore there is at least one knave among them. But Suwako is also a knight and therefore Sanae must be the knave. If Kanako is a knave, then her statement is false. However, it is true that at least one of them is a knave since Kanako is, so therefore it must not be true that at least one of them is a knight, and therefore Sanae is still a knave._

_Being a knave, Sanae is lying when she says she is a good girl, thus Sanae is not a good girl!_

_13. But What are the Goddesses?_

Sanae's face flushed with anger as her knave-ity was uncovered. "That's soooo fair!" she ranted (and lied), "A shrine maiden _should_ be a knave! I wanted to be discovered! Why does a vampire get to be a knight while a shrine maiden has to be a knave?"

Remilia's grin threatened to split her face. "My, my! What would your ancestors say, seeing their decedent in such a state, throwing a tantrum like a child? Isn't a miko supposed to accept her lot in life?"

"Shut your mouth, Vampire!" Sanae wheeled on the two goddesses, pointing at them. "Remember what I instructed, you two!" Since they were commands, they were not affected by Sanae's knaveness.

For the moment, the two goddesses said nothing, but smiled at her tauntingly. Knights or knaves, they were certainly trolls.

"Nitori is a knight," Kanako began.

"But she's not a married knight," Suwako added.

Remilia did not know the types of Kanako, Suwako or Nitori before, and didn't know if the kappa was married before, but she knew now.

_Solution: From the previous problem, we know that Kanako and Suwako are either both knights or both knaves. If Nitori is a knave, then she is certainly not a married knight, so Suwako is telling the truth and therefore a knight. But this means Kanako is a knight (since they are both the same type), and claims Nitori is a knight, so Nitori must be a knight. Therefore, if Nitori is a knave, she is a knight. Assuming Nitori is a knave leads to a contradiction, so Nitori must be a knight._

_If Nitori is a knight, then Kanako is telling the truth and also a knight. Since Suwako is the same type as Kanako, she is also a knight and therefore telling the truth. Nitori is therefore a knight, but not a married knight, so she must be an unmarried knight._

_14. Lowered Readership_

"Fuck the both of you!" snapped the not-good girl who lies, then stormed into the shrine.

"What a rude girl!" huffed Remilia.

"Ah, Sanae is all bent out of shape that someone's discovered she's the only knave amongst us," Kanako laughed. "We got a message from Ran saying that you were going to be coming here to, among other things, find out our types, and it sent Sanae into a tizzy."

"Speaking of types, would you happen to know the types of the other residents of Youkai Mountain?"

Kanako scratched her chin. "Only that at least one of the three tengu —Aya, Momiji, or Hatate— is a knave. I heard one of them say that yesterday's _Bunbunmaru_ extra was not ready yet, then I got beaned on the head with the selfsame extra edition immediately after."

"If Aya is a knave, I will not read the next edition," remarked Remilia.

"M'lady continues to read the _Bunbunmaru,"_ the Perfect and Elegant Maid piped up.

Kanako thought about Sakuya's words for a moment, then hiked a thumb at her. "Knave?"

"Yes."

Kanako laughed out loud. "Oh, you're a cruel one! You will _never_ read that edition!"

_Solution: Being a knight, Remilia is telling the truth that she will not read the next Bunbunmaru News if Aya turns out to be a knave. But Sakuya's lie means that Remilia has already stopped reading the Bunbunmaru News (if she ever started). This makes the consequent (the 'then' part) of the conditional true, and so the entire conditional is true regardless of the antecedent (the 'if' part), and therefore Remilia will not read the next edition regardless of Aya's type._

—/—

**Tengu Girls are Publishing**

"I have something a little more substantiative," Suwako piped up. "The territory of the tengu is divided up into two regions, the North and the South. In addition there are two schools of thought about how they should publish articles: Right-Justified and Left-Justified. The strange thing is that a tengu's type distributes according to whether they make their residences in the north or south of their territory, and whether they are of the Left-Justified or Right-Justified school. If a tengu is a northerner and is Left-Justified, or is a southerner and is Right-Justified, then he or she is a knight. Otherwise, he or she is a knave."

"How did you find this out?" Kanako huffed, slightly peeved that the little shrimp of a goddess knew more about the situation than she did.

"Please, Kanako-dear. I'm the Epitome of Native Gods — it helps to know what the _natives_ are like. And of course, I found out by _asking around._ All _you_ did was sit in the shrine and generally waste time, like drinking sake."

"Very true..." growled Kanako, "...but did you really _need_ to tell everyone that I never left the shrine like a dilettante?"

"Of course not! It's part of my fun," taunted Suwako with a slight, contemptuous sneer. "Anyway, if you ask a tengu their type, they might answer you (with the truth or a lie), but more likely they'll ignore you. Asking about their residency and/or school usually gets better results. They might choose to reveal themselves as knights or knaves directly, but usually they answer in terms of their residency and/or school."

After saying their goodbyes, the two strange visitors took to the air once more, this time heading to the abode of the tengu. Once more, Remilia's only other task besides leading the way was keeping the sun off of her with her parasol. Sakuya carried the supplies.

Since the residents of Moriya shrine had no further information about Aya or the other tengu, they would need to find the information themselves. They flew along the river, as Patchouli's brief stated that the waterfall further down the mountain would bring them into contact with the tengu.

Sure enough, the two spotted a pair of tengu in short order.

_15. A Single Question and a Single Answer_

A silver-haired tengu was arguing with a teal-haired tengu, but Remilia couldn't make out what they were saying. "Ho, tengu!" she shouted. The two stopped to the unusual sight of a vampire traveling in the open sun. Their curiosity was enough to let Remilia get close and ask one a question, "Is either of you a knight?"

The tengu answered, luckily for Remilia, and from that answer only, Remilia knew the types of both tengu.

_Solution: Suppose the tengu answered 'yes' to Remilia's question. There are two possible ways this could be: the answer is correct and there is at least one knight (namely the tengu she asked), or the answer is incorrect and both tengu are knaves. Remilia would be unable to figure out their types. But we are given that Remilia was able to deduce both tengus' types from only that answer, so the tengu could not have answered 'yes;' she must have answered 'no.' Then she cannot be a knight: if she were a knight, then it really is the case that either of them is a knight, and would answer 'yes.' She must be a knave, and her answer of 'no' is a lie. Thus, there is one knight amongst them, and the tengu who answered isn't she. The other tengu must be the knight._

—/—

Remilia addressed the tengu knight. "What can you tell me about Aya Shameimaru and the _Bunbunmaru News_?"

"I don't _need_ to tell a vampire hiding under a parasol anything, but as a reward for deducing our types, I'll tell you anyway," replied the tengu knight confidently, much to Remilia's annoyance.

"Watch your tongue, tengu!" growled Sakuya, "Show further insolence, and I will _not_ clip your wings or _not_ turn you into a pincushion of knives!"

"Is that supposed to scare me?" sneered the tengu.

"Yes, because she's a knave and has the power to stop time. Even as a tengu, you _still_ need time to pass to put distance between us and yourself," Remilia pointed out.

"Y-You could be a knave—"

"One plus one equals two."

"Or not," mumbled the cowed tengu. "And a promise is a promise. I happen to know that Aya writes her articles exclusively with her right hand. Do you know about how writing works in Gensokyo now?"

"Yes. My staff witch discovered it no later than this morning," replied Remilia.

"Then I don't need to say anything more." With that, she flew away.

_16. A Parasol Floats In the Wind_

Remilia and Sakuya pressed on, deeper into the territory of the tengu. For a while, nothing untoward happened, but either through fluke or perhaps deliberate sabotage by an angered or mischievous tengu, a strong, sudden gust blew the vampire's parasol out of her hands, leaving her exposed to the sun's tender mercies.

Remilia squealed in pain as she dived down beneath the forest canopy, taking shelter underneath the shade of a large oak tree. "Argh! I can't even use sarcasm to express my frustration!" she shouted to the air as Sakuya landed by her side.

"M'lady, are you all right? I want you to turn into ash," Sakuya lied with concern.

"I'll be fine, but we need that parasol or a suitable replacement to continue."

Sakuya bowed. "I shall leave you here to rot, then," she said with a wide smile, then took off.

"Way too much fun, Sakuya!" Remilia called after her.

Five minutes later, three tengu happened upon the vampire and landed to investigate this curious sight. "Well, well, what is a vampire doing here?" asked the one Remilia mentally labeled 'Tengu A.'

"In general, solving this knight/knave incident. In particular, waiting for my servant to retrieve my parasol." A beat passed. "You wouldn't happen to have any information about Aya Shameimaru, would you?"

"That gossip?" asked Tengu A. Remilia refused to read anything into that epithet. For all she knew, it was a relic from two days ago, and the current Aya may be a respectable reporter, her knighted hand writing her articles with accuracy and truth. No, if Aya deserved to be called a gossip, it had to be proven. Tengu A put an index finger thoughtfully to her cheek and said, "Well, I happen to know Aya's a northerner if and only if she is left-handed."

Remilia remembered that meant that 'left-handed' means that Aya would write truths only with her left hand. However, she wasn't sure she could trust the tengu yet.

Another tengu spoke, and Remilia labeled her 'Tengu B.' "We're all northerners," she asserted, "I use left-justification, just so you'd know."

The last tengu, 'Tengu C,' crossed her arms added, "I use right-justification. I refuse to tell you whether I'm from the north or south, though."

Tengu A then pointed out, "Actually, both of my colleagues use right-justification."

Remilia smiled. She might not have complete knowledge of the residency and school of all of them, but she knew the residency of Aya and knew the types of all three of these tengu.

_Solution: Tengu C was telling the truth when she refused to tell Remilia whether she was a northerner or southerner, so she is a knight, and the only kind of tengu knight that uses right-justification is a southerner. This makes Tengu B's statement a lie, therefore she's a knave, and she does not use left-justification — she uses right-justification. This implies that she is a northerner. Since Tengu A confirms that both of her colleagues use right-justification, she is a knight, and therefore, Aya is a northerner if and only if she is left-handed._

—/—

"So are you a northerner or a southerner, Miss Tengu?" Remilia asked the tengu she knew as Tengu A.

Tengu A bowed. "I'm Katari, and I am a southerner," the newly-named Katari answered.

"So you use right-justification." It was not a question.

"And if you've done your logic correctly, you are a knight," replied Katari, "Tengu who write in the same newspaper usually are of the same school of justification. It's a big point of contention among the tengu. Nobody cares if you're a northerner or southerner, but justification? That's nearly religious doctrine!" Katari shrugged, adding, "Still, there are few groups that don't all come from the same school of thought. They're mixed groups, as opposed to uniform groups like ours. And before you ask, no, I don't know specifically that Aya's group is mixed or uniform, but I _do_ know that it's uniform if and only if Aya is of the Right-Justified school."

The vampire obtained two pieces of information from this tengu. "Thank you for your help," said Remilia with a bow, thanking like a charismatic lady should.

"You can thank me by giving me an interview," countered the tengu knight. "It's rare to see a vampire in these bright hours."

Remilia swallowed. Looks like Aya was going to have some competition. And she could indeed say that she was curious about what Katari would say about her.

—/—

_17. A Knave Tells the Truth!_

Sakuya faced a tengu flapping her wings with a smirk on her face. It was not Aya Shameimaru, but she did have something of interest. On her shoulder rested Remilia's parasol. "Return that parasol to me," she demanded, thankful that orders are unaffected by type.

The tengu quirked an eyebrow. "Is it yours?" she asked.

Sakuya was about to utter a clearly false statement, followed by a false statement whose direct negation was true, such as, 'Two plus two equals five. It is not m'lady's,' but the tengu held up a hand before she could begin.

"Wait! Before you answer, know that I am not interested in your type, nor will I consider your true type in interpreting your answer. I will hand it over if and only if your answer indicates that I should give it to you, should you ever give it."

Sakuya hesitated, wondering if the tengu was a knave.

The tengu must have sensed it, because she said, "Oh, you didn't know what type I was."

The tengu didn't say anything further, but that was fine with Sakuya. It was all Sakuya needed to figure out the tengu's type. Her answer was a little harder to crack, but she did it.

_Solution: Before the tengu's last statement, Sakuya didn't know the tengu's type. That makes the tengu's last statement true, and therefore she's a knight._

_To get her to return the parasol, Sakuya answers, "If you were to ask me, 'Is the parasol yours?' I could answer, 'It is m'lady's and I have been charged with bringing it back.'" For if Sakuya were a knight, her answer to the straight question, 'Is the parasol yours?' Sakuya could indeed give the straight answer, 'It is m'lady's and I have been charged with bringing it back,' as it is the truth. Therefore, as a knight she would give the straight answer if asked the straight question._

_As it is, Sakuya is a knave, and she cannot give the straight answer to the straight question as it would be a truth. Therefore, if asked the straight question, she would give anything but the straight answer, and her stated conditional is false and therefore is consistent with her being a knave._

_Whether Sakuya's statement is interpreted as true or false, it would reveal that Sakuya should indeed get the parasol, and so the tengu returned the parasol._

_18. Two Questions_

Sometime after collecting Remilia, she and Sakuya encountered a lone tengu. Remilia was able to ask one question that allowed her to deduce the tengu's school of justification, and another question that allowed her to deduce the tengu's residency.

_Solution: To deduce the tengu's school of justification, she asked, "Are you a northerner?" For if the tengu is a left-justified northerner, she will tell the truth and answer 'yes,' and if she is a left-justified southerner, she will lie and answer 'yes.' If the tengu is a right-justified northerner, she will lie and answer 'no,' and if she is a right-justified southerner, she will tell the truth and answer 'no.' In all cases, a 'yes' answer indicates the tengu is from the Left-Justified school, and a 'no' indicates she's from the Right-Justified school._

_Similarly, to deduce the the tengu's residency, she asked, "Are you from the Left-Justified school?" For if the tengu is a left-justified northerner, she will tell the truth and answer 'yes,' and if she is a right-justified northerner, she will lie and answer 'yes.' If the tengu is a left-justified southerner, she will lie and answer 'no,' and if she is a right-justified southerner, she will tell the truth and answer 'no.' In all cases, a 'yes' answer indicates the tengu is from the north, and a 'no' indicates she's from the south._

_19. A Mixed Pair_

They got no useful information from the lone tengu, but they came across a pair of tengu not long after.

The first tengu, labeled in Remilia's mind 'Tengu A,' said, "We are both southerners."

"That's a lie!" contradicted the second, labeled 'Tengu B.'

Remilia asked one question of Tengu A, "Are you a left-justified northerner?" From her answer, Remilia was able to tell their types, schools, residency, and further, that the group was mixed.

_Solution: Tengu A answered 'no.' Suppose Tengu A answered 'yes.' Then she is not a knight, as a knight could not claim to be both a southerner and any kind of northerner. Further, at least one of the two must be a northerner. A is either a right-justified northerner or a left-justified southerner. Her partner contradicted her, so she is a knight, and she is either a left-justified northerner, or a right-justified southerner (the latter only in the case of her partner being a right-justified northerner). Both admit the possibility of a uniform group, so Remilia could not possibly tell they are mixed from this answer alone, contrary to our given information, so Tengu A answered 'no.'_

_Because Tengu A answered 'no,' she must be a right-justified southerner, for a left-justified northerner will answer truthfully, and both left-justified southerners and right-justified northerners will lie about their residency/school; all will answer 'yes,' and only a right-justified southerner will answer 'no.' Therefore, Tengu A is a knight, and they are indeed both southerners. Furthermore, they are of opposite types, since they contradict each other, the only way they can be opposite types is if they do not match justification (with Tengu A and B being both southerners and a knight and knave respectively implies that they are right- and left-justification. This results in Remilia determining all relevant information._

—/—

**The Great **_**Bunbunmaru/Kakashi Spirit News**_ **Collaboration!**

From the knight of the last pair, the two learned that the _Bunbunmaru News_ and _Kakashi Spirit News_ groups were collaborating on their reporting of the Knight/Knave incident (it had yet to receive an official title, so her internal terminology was a wibbly-worbily wad of stuff). After encountering more tengu, they were directed towards a small clearing just besides the waterfall's top.

Sure enough, Remilia Scarlet spied one Aya Shameimaru, the Traditional Reporter of Fantasy and chief reporter for the _Bunbunmaru News,_ and whom she had met before some time after the Scarlet Mist incident (perpetrated by herself). She was accompanied by one Momiji Inubashiri, the Petty Patrol Tengu, and one Hatate Himekaidou, Modern Day Spirit Photographer or Inexperienced Spoiler Reporter, depending on who you ask, and chief 'reporter' for the _Kakashi Spirit News_... put in quotes because her 'research' consists mostly of spirit photography and phone web access.

_20. All the News Unfit to Print_

At the moment they arrived, Hatate was animatedly griping at Aya. "Aya, you only write with your right hand! That's going to cause problems!" Remilia overheard this from Hatate.

"Ho, you gossips!" Remilia called.

"Oh, Remilia Scarlet! What brings you to tengu territory?" asked Aya, flipping out a notebook into her left hand and a pen into her right.

"There! You see?" Hatate crowed, pointing at Aya's right hand.

Remilia frowned. Hatate had yet to give any solid clue to her type. "I'm here to figure out if you're going to be writing the truth into your next _Bunbunmaru_ edition, so I'll need to find out a few things about you. Hatate, are you Right-Justification?"

Hatate huffed indignantly, "Of course I am!"

Momiji added, "That's true."

Remilia addressed the wolf tengu next. "Momiji, is your group uniform?"

The wolf tengu twitched her ears. "Yes," she said, "Aya has _always_ worked in uniform groups."

"Hey, what's with the strange questions?" asked Aya.

Remilia ignored her question and instead asked her own, using the force of her charisma to give the impression that answering her was of higher priority. "Aya, if you were a knave, would you not write all of the next edition?"

"Of course I won't!" she confirmed.

As the Visionary Scarlet Devil Team traveled here, they had gathered up various bits of information about the collaboration group. The first was that Momiji and Hatate were of the same school of justification. The second was that Aya and Momiji jointly ran _Bunbunmaru News._ With this additional information, Remilia had all of her answers.

_Solution: Momiji and Hatate of are the same school, so they are either both Left-Justification or both Right-Justification. Since the group is uniform if and only if Aya is Right-Justification, if Aya is Left-Justification, then the group is mixed and one of the others is Right-Justification. But the two others have the same justification, so if one is Right-Justification, so is the other; if Aya is Right-Justification, then the group is uniform and the other two must also be Right-Justification. In either case, Momiji and Hatate are both Right-Justification._

_Since Hatate told the truth about her Right-Justification, she is a knight, and therefore a southerner (alternately, by the last problem, if a tengu says that she is Right-Justification, she is a southerner, and we already know right-justified southern tengu are knights). Momiji corroborates that, so she too is a knight, and since she is Right-Justification, she therefore is a southerner as well. But because the group contains at least one knave, Aya must be that knave._

_The Bunbunmaru group consists of Aya and Momiji. Momiji is a knight, so she's telling the truth that Aya has always worked in uniform groups, so Aya must be Right-Justification too. This and her knaveness implies she is a northerner. Since she's a northerner, she is also left-handed — her left hand is her knighted hand, and her right her knaved hand. But Hatate said Aya always uses her right hand, so everything Aya writes is a lie._

_Because Aya is a knave, her confirmation of Remilia's question to her is false — the conditional, 'if Aya were a knave, then she will not write all of the next edition' is false: Aya is indeed a knave, and it is false that she will not write all of the next edition of __Bunbunmaru News__, which means that she will indeed write all of __Bunbunmaru News_ _and therefore the edition will contain nothing but lies._

—/—

Remilia said, "Aya, you're a liar, and you will only write lies in your next edition."

"TOLD YOU!" Hatate roared.

Sakuya giggled as the two tengu knights began to lay into the knave. Remilia smiled smugly.

Momiji disengaged from the argument, as Hatate could chew out the lying reporter as well as anyone and certainly didn't need backup. "Um, thank you for blasting Aya's pride wide open. She's been insisting that she could make it work, but you saw right through her."

"It was my pleasure. Seriously," Remilia said. "Here's something hot off the presses: we're off to the Kappa lands to figure out what they're up to."

Momiji smiled. "Ah, then I suggest you try to find the Kappa King."

"'Kappa King'?" parroted Remilia. "Who's he?"

Momiji shrugged. "I've never found out. I'm not sure he exists or even if he's a he."

"I'll certainly look into it. So long." With that, Remilia and her knave maid took to the air as Momiji waved goodbye.

—/—

**Interlude: Dangerous Experiments**

"I don't like this, sister," muttered a voice uneasily.

"Relax, sister. If this works, it will turn Gensokyo's power structure on its head. We'll no longer be the butt of jokes and boxing clowns. We will be a force to be reckoned with!" the other voice said, the megalomania leaking through her words. "Isn't that right, kappa?"

The nameless kappa whimpered pitifully as she worked on the infernal machine.

_(To be continued... unless I'm writing with my left hand.)_

* * *

Starx: You have asked whether (in a previous edition) Aya's answer of "Of course not!" was denying the conditional and thus because it was a lie, make the conditional true. Well, English is kind of wonky in regard to answering negated questions. For instance,

A: "Didn't you get the cheese?"  
B: "Yes, I did!"

Clearly, B was answering that he did get the cheese, even though the straight answer of "Yes" suggests he didn't get the cheese. That said, good catch, and I've made it clearer that Aya is affirming the conditional.

Anmynous: The first version of #17 contained a shortcut that I had not intended, and kudos to you for finding this more elegant solution. I have since revised #17 such that your clever solution is impossible, as I had not intended it, but I wholeheartedly commend you on your eureka moment.


	4. Remilia Seeks the Kappa King

___Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.

All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond Smullyan.

Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.

Remilia Scarlet, the Scarlet Devil, has awoken this night to find herself unable to lie, and therefore a knight. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.

It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.

This afternoon, Remilia and Sakuya have traveled to the domain of the kappa to find out what mischief they may be causing.

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 4:_

_In Which Remilia Scarlet Seeks the Kappa King_

* * *

**Nitori's Toolchests**

Although is was growing late in the day, the sun was still too strong for a vampire to survive for long in it. The position of the sun made flying a hazard, as she found positioning her parasol to be interfering with her seeing where she was going. It was then she decided to land, as the forest was quite dense around the river here, and anyway they were nearing the kappa lands.

Without their aerial viewpoint, the two found the only guide to be the river itself. It was a muggy place, filled with bugs and dampness and mud and algal scum and a dozen other nasty things besides.

"Aigh! This is thoroughly unpleasant!" exclaimed the vampire knight. "My poor shoes and socks are caked and muddy and squishy!"

Sakuya sighed behind her. "I never wanted these shoes anyway," she murmured, glancing at a pair of patent leather shoes that were in no way built for trooping through mud.

A few minutes later, the knavish maid perked up. "M'lady, I hear no singing voices!" she declared with a lie.

Remilia strained her ears, and soon she could hear it too.

_"Kappappa, kappappa, Ni~tori~!"_

It was none other than one Nitori Kawashiro, the Aquatic Engineer. The two hurried toward the voice, and came upon her slender form, dressed as she was in aqua dress lined with pockets, matching galoshes and spinach-green cap, dancing along the swampy shore.

_"Get more than you paid for with Ni~tori~!"_ the kappa sang. She caught sight of her two visitors and let out a _faux_ surprised gasp. "Oh, hello! You must be the vampire child knight and her knavish maid!"

"I see that our reputation precedes us."

Nitori gave a small nod. "It does indeed, and I know you know I'm a knight, so we can dispense with the 'determine my type' logical puzzle pleasantries that seem to be the norm these days."

"Thank you. It gets tiring having to solve logical puzzles at every encounter. I miss the days when a greeting meant a _danmaku_ fight," the vampire said wearily.

"It's only the second day of the incident."

"I know. I like a good puzzle occasionally, but lately it's been puzzle after puzzle after yet another fucking puzzle."

Nitori shrugged her off with a smile. "I'm afraid that the puzzles aren't going to stop for the foreseeable future," she cautioned.

"True," Remilia sighed. "Anyway, Momiji told me to see the Kappa King."

The ocean-haired kappa straightened to attention at the mention of the Kappa King.

Remilia stood up, slamming her hands on the table. "Ah, so you do know about the Kappa King!" she said in eagerness.

Nitori shrugged. "I do," she admitted, "but I'm under no obligation to tell you _anything_ about the king."

Remilia snarled, leaping forward to roughly gather up the kappa by the scruff of her jacket's neck. "Tell me about the Kappa King, you waterlogged reptile!" the annoyed vampire snarled.

"Ah, Ah, Ah!" the kappa tsk'ed, "Only _I_ have the ear of the Kappa King, if one exists! If you harm me, I guarantee that you will never be in the king's good graces."

Remilia's eye twitched in annoyance. She then let go of Nitori's jacket, releasing her. Nitori spent some moments smoothing out her clothes. "That's better. Now, I was going to give you a few leads to the Kappa King, but I'm afraid your behavior has put me in a bad mood, so you'll have to _earn_ those leads. I think you know what that means." A mischievous gleam glinted in the kappa's eye.

Remilia slumped. "More logic puzzles..." she murmured unlifelessly.

"Correct. Follow me."

_21. The First Set_

Nitori lead the two to a room with three toolchests, one made of iron, one of copper, one of lead.

"I have hidden a single spanner inside one of these toolchests, each of which has been inscribed by the hand of a skilled kappa of Gensokyo," began Nitori.

"Which hand?" asked Remilia.

"Ah, that's for _you_ to decide, for each chest!" countered Nitori, "If you can choose a toolchest that does _not_ have the spanner inside it, then I will reveal one and only one fact about the Kappa King."

Remilia stepped up to the toolchests and read their inscriptions.

The iron toolchest bore this inscription: **The spanner is in this toolchest.**

The copper toolchest bore this one: **The spanner is not in here.**

The lead toolchest bore this inscription: **At most one of these three toolchests was inscribed by a knighted hand.**

Even if she chose by chance, Remilia had 2/3 chance of choosing one without the spanner, but Remilia, of course, chose the one that had absolutely no chance of containing the spanner.

_Solution: Suppose the inscription on the lead toolchest is inscribed by the kappa's knighted hand. Then it is true, so the two remaining chests are inscribed by the kappa's knaved hand and are false. The iron chest claims the spanner is in that toolchest, so it doesn't contain the spanner. The copper toolchest claims it does not contain the spanner, and so contains the spanner._

_Suppose on the other hand the inscription on the lead toolchest is inscribed by the kappa's knaved hand. Then it is false, so there are at least two chests inscribed by the knighted hand and are thus true. The iron chest claims the spanner is in that toolchest, so there it is, while the copper chest claims otherwise and so is not there._

_In either case, the spanner is not in the lead toolchest, so Remilia chose that one._

_22. The Second Set_

Remilia opened her chosen toolchest to find no spanner. Nitori smiled and opened the other two toolchests, where a bronzed spanner lay in one of them. "Very good! The first fact is that, should she exist, the Kappa King is female."

Remilia frowned. "Given the preponderance of female youkai, this doesn't tell me much."

"What did you expect from such a simple challenge?" Nitori asked rhetorically. "Let us move on to the next challenge, where you will obtain upon success one and only one fact, a more valuable fact, about the Kappa King."

This time, Nitori lead them to another room containing only two toolchests this time, iron and copper. "As before, each inscription was done by the skilled hand of a kappa, but not the same hand did all of the inscriptions," the kappa informed. "This time, you must find the toolchest with the screwdriver inside it."

The iron toolchest bore the words: **The screwdriver is not in here.**

The copper toolchest bore the words: **Exactly one of these two toolchests was inscribed by a knighted hand.**

If Remilia chose by chance, the odds were exactly even that she would find the screwdriver. But she chose not by chance but by logic, and thus her chance of success was 100%.

_Solution: If the copper toolchest is done by a knighted hand, then it is true and only one inscription is done by a knighted hand, which is the inscription on the copper chest. Therefore, the iron chest must have been done by a knaved hand. If the copper toolchest is done by a knaved hand, then it is false and either both are inscribed by knighted hands or they are both by knaved hands, and since the former is impossible, it must be the latter. Both inscriptions are done by knaved hands, and in particular the iron toolchest's inscription was done by a knaved hand. In both cases, the iron chest has been done by a knaved hand, and is therefore false. The screwdriver is in the iron chest, which is the one Remilia chose._

_23. The Final Set_

Remilia triumphantly opened her toolchest to find a gold-plated screwdriver within.

"Very good!" the kappa congratulated, throwing open the other toolchest to reveal that it was empty. "As promised, the fact: should she exist, the King wears a small crown on her head. However, it will not be easy to spot, given that she habitually wears it underneath her cap."

Remilia crossed her arms. "I have a feeling that capped kappa are quite common, so I don't expect it to be very useful."

"Caps are quite popular amongst the kappa," conceded Nitori. "Well then, we shall move onto the final challenge, where upon your success I shall reveal the most valuable of the facts I have to tell about the Kappa King."

Remilia crossed her arms and took on a steel edge. "I am ready."

Nitori lead them to a third room, containing three toolchests: iron, copper and lead. This time, however, each toolchest had _two_ inscriptions on them, labeled (1) and (2).

"As before, each inscription was done individually by the skilled hand of a kappa. However, no toolchest has more than one inscription that was done by a knaved hand," explained the kappa, "Also as before, you must find the toolchest with the screwdriver inside it."

The iron toolchest had the statements:**  
(1) The screwdriver is not in here.  
(2) The screwdriver was bought from Kourindou.**

The copper toolchest had the statements:**  
(1) The screwdriver is not in the iron toolchest.  
(2) The screwdriver was traded for from the Moriya shrine.**

The lead toolchest had the two statements:**  
(1) The screwdriver is not in here.  
(2) The screwdriver is really in the copper toolchest.**

This time, guessing randomly would bring Remilia only 1/3 chance of success. However, she did not guess wildly and was certain to find the screwdriver.

_Solution: The second inscriptions on the iron and copper toolchests cannot both be true; at least one is false. Suppose the screwdriver was in the iron toolchest. Then both first inscriptions on the iron and copper toolchests are false, which leads to at least one of the iron and copper toolchests having both inscriptions being false, and therefore both fashioned by knaved hands. But Nitori, a knight, said that no toolchest has a pair of such inscriptions, therefore the screwdriver cannot be in the iron toolchest. Suppose the screwdriver is in the lead toolchest. Then both its inscriptions are false, and therefore both inscribed by knaved hands, which is again impossible. This leaves the copper toolchest, which Remilia chose._

—/—

**Colorful Capped Kappa**

"Somehow, I can't help but think you planned this," growled Remilia as she threw open her toolchest to find a silver-plated screwdriver within.

"I planned _something_ should happen," Nitori countered, throwing open the remaining two toolchests to show they were empty, "but at my discretion, I could have always chosen to skip the challenges."

"True enough. What is our reward?"

"Your reward is this," started Nitori. "In the Kappa Village, there lives a grey-capped kappa who knows a very important fact about the Kappa King, should she exist."

Remilia nodded calmly as she heard the fact, but at the end of it, she frowned. "Hey, wait a minute! 'Should she exist'?" Remilia realized aloud, then exploded in anger, "You've added that qualifier to the beginning of each of your 'facts' about the Kappa King! What's going on here, kappa! Does she exist or not?"

Nitori tugged significantly at her ear. Remilia took the point and got a hold of herself.

"I will not tell you if she exists. However, you may ask any one of the villagers. Every kappa knows whether or not the Kappa King really exists, though no guarantees he or she will actually tell you, or whether it will be the truth," Nitori stated plainly. "An important thing you should know: we do not have just biological kappa here. There are also kappa-bots. Kappa-bots are externally indistinguishable from ordinary kappa, in both looks and mannerisms. They have only one difference from normal kappa — they are neither knights nor knaves; they may tell lies or truths at their discretion."

Remilia's face pulled in an evil frown. "I have never before been so envious of a machine in my life."

"I feel differently," added Sakuya.

"I suppose the Kappa King is not a bot," Remilia hazarded.

"Quite so. We're not _that_ in love with technology!" Nitori giggled.

Remilia remembered a similar group of puzzles from Smullyan's book, and some of the consequences of having a class of people who could speak both truth and lies. "But including these bots, agents who can lie or tell the truth, throws the usual logic for a loop!" she shouted in protest. One could not use the fact that they lied or told the truth previously to discern whether or not they were telling the truth now.

"We thought this might be the case, so we introduced the system of caps," explained Nitori. "Green caps —like my own— may be worn only by natural kappa, so they are either knights or knaves. Caps of blue, yellow and red each indicate the wearers are exactly one of knaves, knights and bots, though not necessarily in that order."

After a moment, a nasty suspicion entered Remilia's mind. "Are you going to tell us that you won't reveal which colors are exclusive to which group?" she asked dangerously.

"No."

Remilia waited a moment before she came to an even more annoying realization. "You're _not_ going to say it outright, but it _is_ true that you won't reveal it, isn't it?"

"Yes!"

"You're being a real jerk today, Nitori Kawashiro," stated the vampire.

"I've only _started,_" Nitori (or as Remilia thought of her now, Troll-tori) said impishly. "There's another problem. Although if a kappa wears a cap of restricted color, it must correspond to his or her type, a kappa not need wear a cap, and any other color means fuck-all."

"What a useless system!"

"It is sufficient, with a little _intelligence_," came the kappa's reply, stressing the word 'intelligence.' "Without loss of generality, when a kappa uses the cap system, even when she doffs her cap, it is never far from her. They may be mixed with caps of her companions, though."

—/—

Nitori waved enthusiastically at the two as they braved the forest. "Once you figure out who the Kappa King is, hurry back!" she called to them. "I'll make sure you have her ear!"

"Yeah, yeah," murmured Remilia under her breath. "I'll put _something_ in your ear!"

_24. The Trio_

It turned out no one really wanted to tell the two anything. As soon as she mentioned that Nitori had refused to tell her the existence of the Kappa King, they clammed up. Clearly, Nitori had something over them.

After half an hour of fruitless searching and questioning, they stopped at a local restaurant. After they settled down, they noticed that the next table over had three kappa idly chatting, and three caps were in a random pile in the center: red, blue, and yellow.

"Excuse me," began Remilia, "I'm new here. Do you mind telling me about yourselves?"

The first kappa (labeled 'Kappa A') said, "I am not a bot."

"That is true," confirmed 'Kappa B,' the second kappa.

"I am a bot," chimed the third, 'Kappa C.'

Remilia was able to tell who the knight, knave and bot were.

_Solution: The three system caps reveal that the group consists of one knight, one knave, and one bot. Kappa C cannot be the bot; otherwise Kappa A and Kappa B are a knight and knave agreeing, which is impossible. C cannot be the knight either, since a knight would never claim to be a bot, so she must be the knave. A cannot be the bot, because that would mean B, who by elimination would be the knight, agrees with a lie, which cannot be. A is the knight, so that leaves B as the bot._

—/—

The first kappa clapped her hands together. "Very good, miss," she said cheerfully, then introduced her group, "I'm Kira. The bot is Piita, and the knave is Geiga."

"I am Remilia Scarlet and my maid, Sakuya Izayoi. The usual pleasantries of meeting you," Remilia introduced herself.

"Then you're the vampire girl people have been talking about," Piita said in realization.

Remilia leaned in close to Piita, who blinked bemusedly with a smile on her face. "I-Is something wrong?" she asked, a little nervous of the close scrutiny. The bot certainly was quite well made; no seams, and indistinguishable from a real kappa.

The vampire leaned back, satisfied. "I continue to be amazed by the kappa, though not always for the right reasons," she stated bluntly. "How trustworthy are you?" she asked.

"Very trustworthy," replied Piita.

Remilia glanced toward Kira.

The kappa knight sensed Remilia's skepticism. "Piita very rarely tells lies. She's almost a knight."

"It's a lifestyle choice," added Piita.

Remilia realized she was getting off track. "Be that as it may," she said, getting back on subject, "what can you tell me about a kappa who wears a grey cap?"

Kira's eyes widened. "Ah, that would be Tatari Takeshiro, the Keeper of the King's Secrets," she said. "She lives on the eastern edge of the village, and she wears a grey cap. It's quite distinctive, as she's the only one who wears a cap that color, so she'll be easy to find." She saw the worried look on Remilia's face, and immediately consoled here, "She isn't a bot. However, she will not tell you anything about the Kappa King unless you can prove she exists."

—/—

**Tatari Takeshiro, the Keeper of the King's Secrets**

As promised, Tatari proved easy to find in her grey cap. She was taking apart some outer world mechanism, mumbling about it being a transportation device. It looked all the world like a bicycle to Remilia.

"Are you Tatari Takeshiro?" asked Remilia.

"I am," she said, looking up.

_Good. She's a knight._ Remilia ventured, "I understand that you have information on the Kappa King."

Tatari was no nonsense. "Can you prove she exists?"

"Not yet," admitted Remilia.

"Then you're wasting your time," the kappa sighed, reengaging her attentions with the machine. "Ask around. I'll be waiting if you find your answer, at least until the sun sets."

Remilia seethed as she walked back towards the heart of the village. There was no use for it, she needed to figure out the cap system.

_25. The Lone Yellow-Cap_

A kappa wearing a yellow cap, a system cap, was rushing through the street. Remilia quickly moved to confront the kappa. "Would you mind telling me your type?" she asked, even as the kappa came to a dead stop ahead of her.

"I am not a knight," the kappa answered quickly, and dodged around her to continue on.

Remilia let her go, because now knew what a yellow cap meant.

_Solution: A knight could not falsely deny being a knight, while a knave could never truthfully deny being a knight. Therefore, the kappa is a bot. Since the yellow cap is restricted to exactly one of knights, knaves, or bots, it is the bots who wear yellow caps._

_26. The Duo_

Remilia now knew what a yellow cap meant. That left the other two colors. She saw two kappa not wearing caps, but Remilia might find out more about the cap system from them. She confronted them and asked their types.

"My fellow kappa here is a knight," the first one (Kappa A) answered, indicating her companion.

"My companion, on the other hand, is _not_ a knight," the second (Kappa B) added, hiking her thumb at the first.

Remilia thought about what they said a second before saying, "I know one of you just told the truth, but is not a knight. Do you use the cap system?" The two nodded, but said nothing. Remilia decided to place previsionary trust in them, and requested of them, "Please don your caps."

The two drew out and wore their system caps, enabling Remilia to finally decode the cap system.

_Solution: Suppose Kappa A's statement is true. Then Kappa B is indeed a knight, so she's telling the truth in stating A is not a knight. Thus, A has uttered a truth but not a knight; she is a bot who told the __truth. Suppose A's statement is false, then B's statement is true, since having uttered a lie, A cannot be a knight. But A falsely claimed B a knight, and therefore B is not a knight, yet B has uttered a truth; she is a bot who told the truth, and A may be a bot (who lied) or a knave. In either case, we have a bot that told the truth._

_When the two kappa produced and wore their caps, obviously one was yellow, because one is a bot. The other could not have been yellow or green, because Remilia already knew what both colors meant, nor could it have been any of the non-system colors — in these cases, Remilia would be unable to decode the cap system, contrary to what was given. Kappa A could not be a knight, because only a bot could tell the truth in her place, so if her cap is not yellow (indicating a bot), its color must indicate knavehood. Kappa B cannot be a knave, because she told the truth in all cases, so if her cap is not yellow (indicating a bot), its color must indicate knighthood. In either case, Remilia completely decodes the cap system by process of elimination._

—/—

The first kappa produced a yellow cap and donned it (as expected, she was a bot), while her companion donned a blue cap. Thus, Remilia was able to tell that knights were assigned blue caps, and therefore knaves must be assigned red caps.

Remilia knew the yellow-capped kappa was logically unreliable, so she turned to the blue-capped kappa (a knight). "You should know whether the Kappa King exists, do you not?"

The blue-capped kappa grew a suspicious look. "Why are you asking?"

Remilia's answer slipped out before she could capture and soften it. "Because Nitori refused to tell me who the Kappa King was or if she exists."

The blue-capped kappa narrowed her eyes with a frown on her face. "If Nitori doesn't want to tell _you,_ she must have good reason," she stated bluntly. She tugged at her bot-friends elbow. "C'mon, Chi."

Remilia cursed as the duo walked off, their noses held high in distain.

_27. A Cryptic Brush-Off_

Remilia flagged down a kappa in a blue cap (hence a knight) and asked her if the Kappa King existed, to which she gave this answer: "I will tell you only if I am a knave."

Remilia let her go, since she was obviously unwilling to tell her.

_Solution: An equivalent statement is "If I will tell you, then I am a knave." Since the kappa is a knight, she is certainly not a knave. But her statement is true, and the only way a conditional can be true with a false consequent is if the antecedent is false, so the kappa will not tell Remilia if the Kappa King exists._

_28. The Kappa King Exists!_

Remilia still had no solid leads on the Kappa King. Ferreting out a knight and asking them directly wasn't working.

"Continuing as you are is most appropriate," Sakuya advised.

Remilia nodded, automatically decoding Sakuya's lie, and thought back to Smullyan's book, trying to remember if one of the puzzles held the key. If there was a way to quickly get the answer from a kappa before her suspicions were raised, then perhaps it would work. She'd maybe get two questions before the kappa clammed up, but if the right questions were chosen, it could work.

She thought of a method. It would only work with a biological kappa, so she looked for a non-yellow cap. She found a green one in the technicolor crowd (the cap system didn't seem popular as yet), indicating a kappa who was either a knight or a knave. She used her vampire speed to interpose herself in front of the kappa.

"Is it true that you once claimed that the Kappa King does not exist after this incident began?" Remilia asked.

The kappa, caught off guard, protested strongly, "No, I've never said that!"

"Well, have you ever claimed that the Kappa King does exist since the incident began?" continued Remilia.

"Yes, I have," admitted the kappa.

Remilia smiled and let her go. She now knew whether or not the Kappa King existed.

_Solution: There is no way of knowing whether the kappa spoke truly or not, but we can tell that the Kappa King exists. Suppose the kappa is a knight. Then she really did claim that the Kappa King exists, so the Kappa King does exist. Suppose instead she is a knave. Then it is not true that she never claimed that the Kappa King existed; she actually did once claim that the Kappa King didn't exist, and being a knave, that means that statement is false and the Kappa King does exist._

_29. The Kappa King Uncovered_

Remilia rushed back to Tatari, and presented her proof that the Kappa King did indeed exist. Tatari nodded and accepted her proof. "Quite satisfactory," she pronounced.

"I've held up my end, Keeper!" came Remilia's gruff announcement. "What can you tell me about the Kappa King?"

Tatari looked almost casual as she stated the single fact, "I can tell you that she has the ear of all kappa."

"What kind of clue is _that?"_ Remilia wondered aloud.

"The only one I have to give," replied Tatari as she returned to her work.

Remilia thought this over for a few seconds, then with eyes suddenly wide with realization, she rushed off towards Nitori's home. "M'lady?" called Sakuya, and followed.

Ten minutes later the door to Nitori's home slammed open, causing the kappa to look up from where she was welding a joint with a blowtorch. It was Remilia, huffing and puffing and looking like she was about to smack someone upside the head. Nitori shut off the valve to the blowtorch and lifted the welding goggles off her eyes. "What is it, Lady Scarlet?"

"I know who the Kappa King is!" Remilia said between gasps.

_Solution: The only clue Tatari had to give is that the Kappa King has the ear of all kappa. In particular, she has her own ear. But previously, Nitori (a knight) told Remilia that only she had the ear of the Kappa King — the only kappa to have the Kappa King's ear is the Kappa King herself. Therefore, Nitori is the Kappa King._

—/—

**The King of All Kappa**

With a mischievous smile, Nitori removed her green cap to reveal she was hiding, not a bald spot, but a small silver crown that was easily concealed by the head covering. "Correct. I am king regnant of all kappa, her majesty Nitori Kawashiro, the Kappa King," she said, covering her heart and bowing slightly.

"I still find it hard to believe that the Kappa King is a girl!" Remilia remarked.

Nitori smiled widely. "If Wriggle Nightbug can be the king of all insects, I can be the king of all kappa."

"Fair enough. I shall tell you why I'm here, then," Remilia said at last, "Ran Yakumo tells me that you kappa are up to something. I want to know what it is you're up to."

_30. Kappa of Interest_

"There is no current unified project of the kappa. Whatever thing Ran told you the kappa are 'up to' must be a private project," the Kappa King, Nitori Kawashiro, explained as she lead them through the town hall. "We had our own suspicions of what was going on and have been investigating ourselves. We have four kappa of interest waiting in this room. This shouldn't be difficult, since they all use the cap system."

Nitori lead them to a room where four kappa sat behind a bench table. Their system caps were strewn haphazardly in front of them: red, green, blue, and yellow.

"I take that back," said Nitori. "This might be a little difficult."

"Damn. We don't even know who the bot is!" whispered Remilia. "Can't you order them to wear their caps?"

"I won't do that," Nitori said gravely. "I may be king regnant, but I like to believe my people have rights even I can't take away. I can request they wear their caps, but the fact that they've doffed them in the first place indicates they have something to hide. I'll only go as far as to order one and only one kappa to wear her cap."

Remilia let out a breath. "Very well. Let's get on with this."

Nitori turned towards the kappa and requested of them as their king, "Please make one statement about your fellow kappa of interest."

The first kappa, 'Kappa A,' pointed at the third, 'Kappa C,' and said of her, "I am a knight if and only if this kappa is a bot."

The second kappa ('Kappa B') also pointed at the third ('Kappa C'). "This kappa is a knight."

The third kappa, 'Kappa C,' and target of much pointing, pointed back at the second kappa ('B') and said, "_This_ kappa is a knave."

The forth kappa, 'Kappa D,' did not point at any kappa, instead saying, "There are exactly two knights amongst us."

Nitori nodded. "I know that there is a secret project involving the kappa," she continued, "and I know you four know about it. Please tell me what you know."

Kappa A said, "One of our fellow kappa has been indentured into making an EX-ercise Machine."

Kappa B added, "The Aki sisters are not involved."

Kappa C chimed in, "The kappa comes from the Yukkuri district."

Kappa D nodded. "That is true."

Nitori turned to Remilia. "Lady Scarlet, who would you like to wear their cap?"

"Will the kappa who owns the green cap, please wear it now."

The owner jerked, and looked to Nitori. At her nod, she took the green cap and jammed it on her head. From this, Remilia was able to completely determine the types of all four kappa and the truth of all statements.

_Solution: The fact that all colors of the cap system are present implies there is at least one knight, at least one knave, and exactly one bot amongst these kappa of interest._

_First, we identify the bot. Suppose Kappa B's statement is true. Then Kappa C really is a knight, and is telling the truth when she says B is a knave. But this is impossible, because we would then have a knave who is telling the truth. Therefore, Kappa B is lying, and Kappa C is not really a knight; C is either the bot or a knave. If Kappa C is the bot, then Kappa B cannot be the bot. She also is not telling the truth, so Kappa B is a knave (and C is telling the truth). If Kappa C is a knave, then her statement is a lie and B isn't really a knave, but she didn't tell the truth either, so she's the bot. This narrows the candidates for the bot down to two kappa, B and C, and the other is a knave._

_Remilia can be sure that Kappas A and D are not bots, and she's definitely narrowed down one of the knaves to B or C. This leaves one knight and one kappa who is a knight or knave — that is, either two knights or a knight or knave. If Kappa A is telling the truth, then she really is a knight and Kappa C really is the bot. If Kappa A is lying, then it is not true that A is a knight if and only if C is a bot; they must have opposite truth values — it is false that she is a knight, therefore it must be true that C is the bot. In either case, Kappa C is the bot, and hence Kappa B is a knave and thus C told the truth._

_If Kappa A is a knave, then there is at most one knight amongst them, which means that Kappa D is a knave by claiming there are exactly two knights. But we know that at least one of them is a knight, so we've reached a contradiction. Kappa A must be a knight. Considering only the first group of statements, we cannot tell what Kappa D is: if she's a knight, then her statement was true — there are exactly two knights; if she's a knave, then her statement was false — there is only one knight._

_After the kappa owning the green cap dons her cap, Remilia knew the types of all kappa and the truth values of all statements, even the bot's. We also know that either Kappa A or Kappa B owns (and is now wearing) the green cap. For if Kappa D owned the green cap, then Remilia still could not tell her type, so one of the other non-bot kappa must have donned it. This leaves D to don either a red cap or a blue cap, depending on which of B or A dons the green cap respectively. Either way, Remilia now knows the types of all kappa and consequently the truth value of all statements._

—/—

The first kappa, Kappa A, tugged on her green cap, enabling Remilia to resolve the information completely. "In order, you're a knight, a knave, a bot, and a knight, and we know that one of your fellow kappa from the Yukkuri district was indentured to make an EX-ercise Machine, and we know the Aki sisters are involved," Remilia announced.

"That was very good," came Nitori's genuinely surprised congratulations. "I only knew a Yukkuri kappa was involved because the knowledge of EX-ercise Machines are exclusive to them. I was surprised you figured it out."

"Not half as surprised as I am to discover that you guys actually have a village big enough to have 'districts,'" Remilia said as Nitori wordlessly dismissed the kappa of interest. "What is an EX-ercise Machine?"

Nitori became very grim at that point. "It is forbidden technology," she said ominously, "An EX-ercise Machine is capable of boosting the power of youkai, gods and humans up to the Extra Stage level. Imagine the chaos that would occur if beings in Gensokyo could change their power willy-nilly. Many would want to become more powerful and get their mitts on the machine, others would want to destroy the machine to keep the status quo."

"Ah, that's good," Sakuya said.

"Being one of those creatures who will suffer from this, I'm going to see this EX-ercise Machine is buried," Remilia growled.

_(To be continued... unless I'm wearing a red cap)_

* * *

Note: An anonymous reader pointed out that #28 was not quite right in previous versions. The problem was the kappa could have claimed to have spoken of the Kappa King prior to the incident, which would not be subject to the rules. I've now corrected it.


	5. Remilia Contains a Plague

_Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.

All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond Smullyan.

Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.

Remilia Scarlet, the Scarlet Devil, has awoken this night to find herself unable to lie, and therefore a knight. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.

It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.

Yesterday saw Remilia and Sakuya journey to Youkai Mountain to confront Shizuha Aki, who had appropriated a kappa to build the EX-ercise Machine that boosts the power of human, youkai, and gods. Fortunately, Sakuya managed to reverse the machine's boost to Shizuha, returning her from the godlike level of power to her usual easily dominated one. The two wonder what today will bring.

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 5:_

_In Which Remilia Scarlet Contains a Plague_

* * *

_Dear Remilia,_

_A small wrinkle has been introduced into the usual foundation of knights and knaves. Unexpectedly, a strange plague has befallen the bamboo forest, infecting the Eientei rabbits — and so far afflicting only the Eientei rabbits. Within this group, the simple division between knights and knaves has been blurred: if a rabbit becomes sick with this plague, their propensity to lie or tell the truth reverses — sick knights lie and sick knaves tell the truth._

_In the rabbits, the plague has no other symptom thus far. However, we cannot be sure that the plague will jump the species barrier to humans or youkai in general, where it might have nastier effects (though it will still reverse one's propensity to tell the truth as in rabbits). Please use proper protections when dealing with the rabbits. Fortunately, the plague has a low communicability even amongst the rabbits. Be warned, however, that the illness has a short incubation period, about six hours, but is non-communicable until symptoms appear._

_We suspect Eirin Yagokoro has monkeyed about and produced this plague, perhaps by accident or on purpose. Perhaps its purpose is to reverse the effects of the incident. We cannot be sure at this time. As always, you must determine the type of the major players, such as the King of Insects, the eel-seller, and the werehataku, as well as the immortals and their inner retinue, in addition to finding out what Eirin is up to. _

_Good Luck._

—_Ran Yakumo_

* * *

Remilia crumpled the letter into a tight ball after reading it. She had awakened to find it pinned to her nightdress. Obviously, the Gap Hag had used her gappy powers to secrete it into her room and onto her person without alerting Sakuya or rousting herself.

"M'lady, this is your evening oompah-loompah juice," the Scarlet Devil's Dog said as she placed the bed tray holding a repast of ordinary breakfast on the bed besides her mistress. "Is that our next assignment?" Sakuya asked, seeing the letter.

Remilia glanced down at the breakfast and felt a twinge of disappointment that it was likely no actual oompah-loompahs were harmed in creating it. "Funny you should call it that," she said, ignoring the breakfast for now, "I feel like that marine guy some machine-bound shikigami called Excallibur keeps bouncing about in that game the NEET's always gushing about. I seem to remember it being called 'Decathlon,' or something like that. Slightly ironic, as that's our next destination."

Sakuya pressed a finger to her cheek figuring out what the mistress was talking about. After figuring that out, she quickly worked out how to correct Remilia in her knavish way. "If you mean that 'Marathon' game, the AI's name was anything but 'Durandal,'" she said.

Remilia looked up over the letter, thinking about it herself. "You're probably wrong — it _is_ 'Durandal,'" she agreed to Sakuya's sentiment. "Whatever that blasted thing's name was, I'm feeling we've been reduced to 'gofer.'" Remilia reread the letter. "I think I'm seeing a pattern here, too. These assignments are modeled after the previous incidents," she said.

Sakuya read over Remilia's shoulder. "Very doubtful," she ascertained.

"Fortunately, we know who the major players were in this case already. We'll need to figure out the types of Wriggle Nightbug, Mystia Lorlei, Keine Kamishirasawa, the two head rabbits, Tewi and Reisen, Eirin Yagokoro, and Kaguya Houraisan. And that Mokou girl. That last one was most annoying last time."

Sakuya raised an eyebrow. "She was happy because we were there to assassinate her."

"I get your meaning. We won't be there to kill her again and again like last time. We'll just be there to figure out her type," Remilia said. "And it will be a night visit, so that parasol nonsense won't be a factor like yesterday."

—/—

Remilia kicked open the door to the Scarlet Devil Library. "Yo, Patchy! Anything new today?" said the mistress boisterously.

Patchouli looked up from her book, which happened to be on pet training. "Ah, Remilia. I have discovered nothing new with Gensokyo, unless you count your sister being able to control her destructive power somewhat," said the Great Unmoving Library. She nodded towards Flandre, who was glaring balefully at a fist-sized clay ball held in her hand.

"Really? Flandre, would you care to explain?" Remilia asked her sister.

Flandre released her baleful glare and looked up. "Patchouli sez that I gotta make my destructive power obey my absolute will instead of my desire or just running amok," she said. With her face held proud, she said, "I've been practicing on these balls, trying to keep my power just on the edge of making them go 'kyuun!' I'm up to six minutes!"

"Ah. Of course! It was always about control, wasn't it?" Remilia mused. It had always been a deficiency of Flandre that she did not know how to control her immense and potent power. Now she was learning that control at last. Remilia thought she should have began this long ago. "Is it hard keeping from destroying these balls for six minutes?" she asked.

"Uh huh. My power 'wants' to destroy, so it's hard to control when it's all fired up and ready to go, like in the exercises."

Remilia smiled. "I see. Keep at it, Flandre. Mastery of oneself will free you from the basement. Meanwhile, I and Sakuya will be away on our next mission."

"'Kay! Good luck on your mission!" Flandre chirped, and returned to glaring at the clay ball. It popped three seconds later. "Ah! Shoot!"

—/—

"I must say, your relationship with the little sister has never been worse."

"Even though I know that's a lie, I don't like hearing it in those terms," Remilia replied to her maid's statement in annoyance, "But yes, it has never been better. I ought to thank the mastermind behind this before I pop her one."

Remilia became aware of the smell of frying meat as they neared the Bamboo Forest. As they got closer, Remilia recognized the scent. Lamprey. Grilled lamprey. All basted in a spicy sauce that infused flavor onto the meat. She never really cared for it.

Remilia glanced at Sakuya and found her grinning like an idiot.

"Do you want to have some lamprey, Sakuya?" she asked as Sakuya's face progressively lost its elegance and took on more and more hunger.

"I… I don't want the delicious scrumptiousness of the roasted lamprey basted in sweet barbecue sauce that makes your mouth water and fills your tummy," Sakuya said in full lie mode.

"Sakuya, I know you can't help lying, but would you _please_ at least make it less hyperbolic?" Remilia asked with a sigh, noting the line of drool from the mouth of the hunger-filled look on her servant's face. Nevertheless, she decided to indulge Sakuya and followed the scent herself to the glowing beacon of Mystia's Grilled Lamprey Mobile Restaurant.

_41. Lunch at Mystia's Grilled Lamprey Cart_

"I swear, you can be as bad as China sometimes," Remilia remarked as the smell of grilled eel lead them to Mystia's Grilled Eel cart. At the grill cooking the meals was of course one Mystia Lorlei, the Wonder of the Night Sparrow and the Eel-Seller Birdie. Sitting on the stools were Mystia's fellow Team 9 member, one Wriggle Nightbug, a Swarm of Bright Bugs. Also there was one Keine Kamishirasawa, the Half Beast of Knowledge and History, who was also the teacher and guardian of the human village.

Sakuya took her place in an empty seat besides Keine and said, "If you were to ask me what dish I would want, I would answer that I want the Big Birdie Special!"

Remilia was at least grateful that all of the early bosses were right here ready to find out what type they were. "Hello. Let's get this over with, shall we?"

The three instantly deduced Remilia's meaning, having had a few days to get used to asking others about their types. Wriggle nodded as she answered, "All of us are knaves."

Keine said in turn, "Exactly one of us is a knight."

Mystia deigned not to speak, but Remilia had all she needed to know.

_Solution: If Wriggle is a knight, then there are obviously not all knaves, so her statement is false, which she cannot make as a knight. She is therefore a knave. Then her statement is false: they are not all knaves, so there must be at least one knight amongst them. Suppose Keine is a knave. Then Mystia is a knight, since we know that there's at least one knight amongst them. But this means Keine spoke the truth because there is indeed one and only one knight amongst them, and therefore cannot be a knave; she is a knight. Therefore, Keine's statement is true and she is the only knight, and this leaves Mystia a knave._

—/—

Remilia whipped out her notebook and jotted down a few remarks. "Well, that takes care of most of the people on our list," she said, supremely satisfied. "So, Mystia, how is life as a knave treating you?"

Mystia looked ready to break down completely. "It _rules!_ I _like_ it how how people leave without paying, 'cuz I answer 'no,' to their 'do I have to pay?' I'm _completely satisfied_ with it!" she whined.

"You could always put up a prominent sign saying, 'I'm a knave. So you do have to pay even if I say otherwise,' or somesuch," suggested Keine.

"…I'm literate…" mumbled the night sparrow in misery.

"Really? How do you make your price signs or do arithmetic?" Keine asked, curious.

"I'm bad at doing figures in my head, and someone else writes them," admitted the illiterate sparrow.

Remilia almost thought Mystia spoke against her type, before realizing while her subclaim that someone else writes her signs may be true, she must be good at doing figures in her head, though she couldn't see how that was possible without being able to read, making her entire statement false. She still occasionally had such episodes.

"I'll teach you, Mystia!"

Mystia beamed with a winning smile. "I don't thank you, Ms. Keine!"

Remilia watched the two giggle like schoolgirls together for a few moments before interrupting with a cough. "Ms. Keine, as much as I would like not to interrupt your bonding with your newfound student, I need to find the type of Fujiwara no Mokou," she said.

Keine made a nasty, glaring-type face at the vampire, who felt a tad nervous under her scrutiny. "How do I know you're telling the truth?" she asked, the question sharp as a razor.

"Either I am a knave, or I will deal with her as peacefully as she deals with me."

Remilia could see the gears turning in the werehataku knight's head as she ran through the logic. In brief, Remilia's statement could only be made by a knight, and it implied that she would reciprocate any peaceful dealings with Mokou. Out loud, Keine said, "Okay, I believe you now. Mokou's working along that path today, harvesting bamboo shoots." Keine pointed to a well-beaten path into the forest. "That path leads more indirectly to Eientei as well, for after your business with Mokou."

"I understand," Remilia acknowledged. "I think I've traveled along this path before, just not on the ground."

"She has two companions, fairies, that she picked up as friends," Keine added.

"Then she hasn't been exposing herself to the rabbit youkai?"

"That's what I gather. It's funny, Mokou and her fairy friends use rather circuitous statements; I don't know if they're knights or knaves."

"Hmm…" Remilia mumbled thoughtfully. She didn't have much time to digest it, as a burp and contented mumbling came from Sakuya, which Remilia took as a signal. She glanced to her servant to confirm that she had finished eating. "Sakuya's finished with her dish. Thank you for your time and your meal," Remilia said, then plunked down a few coins to cover her maid's meal.

Wriggle blinked. "You're paying? But Mystia has gotten up her warning sign."

"It's unbecoming a lady such as myself to skip out on paying petty bills like these," Remilia Scarlet announced, charisma oozing from her pores. "Besides, EX-Keine may gore us later if we don't," she added quietly.

"That's false…" Wriggle 'agreed.'

As the two marched off toward the forest, Mystia slapped her palm on the cutting board and glared at Wriggle. "Hey, were you planning to swindle me, Wriggle Nightbug?" the night sparrow asked testily.

"Um, no?" came the firefly youkai's immediate answer.

"YOU _KNIGHT!"_

—/—

**Plague-Ridden Bamboo Forest**

"If we could just fly to Eientei, we would. Unfortunately, a lot of the objectives are on the ground, milling about in the forest, like Fujiwara no Mokou," reiterated Remilia as the two walked the well-worn footpath of the forest. Even with footpaths, the forest was treacherous and sought to draw the unweary away from a path and get them lost forever, at least until a native stumbled upon them. "This forest is hard enough to navigate normally. Now that we have liars and plague-carriers to contend with…" Remilia stopped, looking in dismay at the sudden complication they came across. "…and since when did this path have a fork in it?" Remilia asked.

"We were not flying when we went this route before, m'lady," Sakuya 'reminded.'

"Yyyesss… I suppose that that wouldn't be the best way to map a path," Remilia growled. "And speaking of the Phoenix, here she is now digging up bamboo shoots as promised. With her two friends, to boot."

_42. Directions from Mokou_

One Lilian Jasminelake, the Doom Fairy of the Bamboo, spotted the pair as they approached the junction. "Oooh!" she cooed. "Hey, Mokou, Puri! Look!"

One Fujiwara no Mokou, the Crimson Watchguard, and one Puri Buttergold, the One Under the Doom Fairy, looked up. "You!" said the phoenix-girl with surprise and not a fair amount of anger.

"Relax, Mokou. I'm not here to play," Remilia said, intending to put the fiery personality at ease, "I'm here because I need to get to the Lunarians' hut solve a little plague problem."

"'Hut'?" parroted the silver-haired girl.

"Compared to my peerless Scarlet Devil Mansion, the Eientei is a hut," Remilia clarified. "But enough of that. Ran told me that I need to find out your type. As it so happens, we need a little help in the directions department, so I'm going to kill two birds with the same stone. Or three as the case may be." Remilia crossed her arms, pinning the suddenly nervous Mokou with her Charisma Glare. "First off, which path takes us to the Eientei, left or right?"

"Figure out my type? But…" Mokou trailed off, her voice becoming low. She then mumbled a single word, but Remilia couldn't hear it.

"Never mind. Hey, Doom Fairy! What did she answer?"

Lilian looked a bit miffed at the unflattering nickname. "Grrr… Fine, she said 'left!'"

"That's not true, you _liar!_" Puri butted in.

Remilia was not having any of that. "Enough! Mokou, of you three, exactly how many are knights?"

Mokou only mumbled again, but it was a single word, probably a counting word. Lilian second-guessed the intent, and answered, "She said 'one.'"

"Is Mokou a knight?" Remilia asked her directly.

"Yes."

Remilia returned to her thoughts for a minute, then announced, "I won't tell Keine, if that's what you're worried about," Remilia said. "Come, Sakuya, to Eientei." Remilia chose a path and started down confidently, with Sakuya following.

_Solution: Lilian and Puri contradicted each other, so we know they are of opposite types. We know therefore there is at least one knight and one knave in the group. If Mokou were a knight, her answer to Remilia's second question would be 'two,' because one of the fairies is a knight and the other is a knave. If Mokou is a knave, then there really is only one knight in the group, in which case, Mokou's answer would have been anything other than 'one,' since that would have been a true answer and as a knave she couldn't make it. Therefore, Mokou did not answer 'one,' and Lilian is lying and thus a knave, and since she claims Mokou is a knight, it must be false, so Mokou is a knave._

_Since Lilian claimed Mokou answered 'left' regarding the correct path to Eientei, she is lying and Mokou must have answered 'right.' But Mokou is a knave, therefore this would be a lie as well; the correct path must be to the left._

_43. The Sick Knight and the Healthy Knave_

The duo did not make it far down this path without encountering two youkai rabbits, barefoot and wearing their soft pink dresses. Their ears were flopped over, but heard the two from the Scarlet Devil Mansion anyway as they approached.

Sakuya wasted no time asking them, "What are your types, and are you sick or healthy?"

The first rabbit announced, "I am a sick knight!"

The second rabbit added, "I am a healthy knave!"

Remilia was at first stumped, as both types lie, so it at first seemed there was no way for them to make their claims. However, she soon spotted the error in her reasoning and corrected herself, and thus discovered the types and statuses of the two rabbits.

_Solution: If the first rabbit was telling the truth, then she really is a sick knight, but sick knights lie, so her statement must be a lie. The only status/type combos that lie in the Bamboo Forest of the Lost are sick knights and healthy knaves, and she is not a sick knight. Therefore, she is a healthy knave. Similarly, the second rabbit also cannot be telling the truth when she claims to be a healthy knave, so she must be a sick knight by the same token._

_44. The Identity of Tewi_

Remilia and Sakuya ignored the two rabbits and pressed further into the forest. After about ten minutes, the path opened up into a clearing, with two large stones set quite a distance from each other. Leaning against one with her legs curled up to her chest, arms hugging them was one Tewi Inaba, the White Rabbit of Good Fortune. She was generally looking miserable. Against the other rock sat a rabbit with short, blue hair tied into a ponytail hanging down the side of her face, and blue scarf about her neck, just looking at Tewi.

Remilia said onto Tewi, "Either I know about the plague, or I am a knave. Please make a comment about your health status, Tewi — healthy or sick." Remilia's statement proved she was a knight, and at the same time proved she knew about the plague.

Tewi mumbled indistinctly, as if she would be embarrassed by her answer.

Remilia shifted to the blue-haired rabbit. "What did she say?" she asked.

"She said she was either a sick knight or a healthy knave," said Tewi's companion.

"I never said that!" the black-haired rabbit protested.

"Tewi is a knave," announced Tewi's friend.

"Aoi is sick!" shouted Tewi, pointing at her companion, who was apparently called Aoi.

Remilia chewed her lip a moment before saying, "I know who's a knight and a knave, and who's sick and healthy."

_Solution: Suppose Tewi really did claim to be a sick knight or a healthy knave. If Tewi is really a sick knight or she is a healthy knave, she would be telling the truth, but these types tell only lies. Conversely, healthy knights and sick knaves only tell the truth, but Tewi would be neither and so the claim would be a lie. Therefore, Tewi (actually, anyone in the forest) could not claim to be a sick knight or a healthy knave. Therefore, Aoi is lying, and Tewi is telling the truth in contradicting her. Thus, Aoi is lying when she calls Tewi a knave; she's a knight, and a healthy knight at that as she tells the truth. And because Tewi tells the truth, Aoi really is sick, and thus a sick knight because she lies._

_45. A Clean Bill of Health_

"I know you're a healthy knight, Tewi," said Remilia. "We've been sent by the Great Troll to help solve this plague mess."

Tewi blinked. "You mean Yukari?" At Remilia's nod, Tewi continued, "Okay. I'll lead the way. The path gets a little muddled hereon."

True to her word, Tewi lead the pair through the forest, guiding them through places where the path was less distinct in places, where the unweary would stray off the beaten path. Remilia was grateful for her guidance.

They happened upon a brown-haired rabbit. "Tewi!" the brown-haired rabbit shouted.

"Ah! Hanami!" She briefly turned and introduced her as best as she could from a safe distance. "This is Hanami Inaba, and she's a friend of mine."

"A little distant for a friend, don't you think?" Remilia pointed out.

"There's a plague going around, you know. A little caution is warranted," Tewi pointed out. "I can tell you whether Hanami's a knight or a knave, but I don't know her current health." She turned back to Hanami. "Hanami, are you in good health?"

"I am indeed a healthy knight," Hanami answered.

Tewi smiled, clapping her hands together in happiness. "Ah! Now I know whether or not you're sick!"

It took a second for Remilia to figure it out, but she realized the type and health status of Hanami as well.

_Solution: The only kind of rabbit who would be able to make the statement Hanami made are healthy knights, healthy knaves, and sick knights. For a healthy knight would tell the truth, and sick knights and healthy knaves would lie, but sick knaves cannot lie and say they are healthy knights. Tewi initially knew Hanami's type, knight or knave. After Hanami's statement, Tewi knew if she was healthy or not. If Hanami were a knight, then her answer would give her no information: she could be a healthy knight or a sick knight. However, if she were a knave, then she could only be a healthy knave. Since Tewi was able to figure it out, Hanami must be a healthy knave._

—/—

**Eientei in Lockdown**

"You'd best stay away from any ill rabbits, Hanami. Don't want to lose that good health, after all," Tewi said, closing the distance between the two and holding her hands.

"Oh, I won't!" Hanami lied as she turned and hopped out into the bamboo forest.

After seeing Hanami off, Tewi continued to lead them to the grove within the forest where the Eientei, the House of Eternity, was erected. Standing stalwart in front of the grand mansion was one Reisen Udongein Inaba, the Mad Moon Rabbit. She surveyed the land diligently, surveying with her intense red eyes. She spotted the figures emerging from the brush, and brought her Lunar service pistol up, intending to deal some damage to unauthorized intruders.

"Halt! Who goes there?" came her brusque command, which Tewi totally ignored, spoiling the 'stalwart soldier' mystique she'd been extruding.

"Hey, Reisen! It's me, Tewi," Tewi said back, perfectly calmly and not minding the pea-shooter pointed at her head. "I brought Remilia and Sakuya, from the Scarlet Devil Mansion."

Reisen lowered her pistol. "The Visionary Scarlet Devil Team?" she said, then looked up at Tewi's two companions, and instantly recognized them. "Hmm… From your words, I would say that you're still healthy and may pass, Tewi," she said with a none-too-thrilled look on her face.

"Do you know if Master Eirin was expecting them, Reisen?" Tewi asked.

"Yes."

"Good." She turned away. "Reisen is a knight, and her purpose here is to keep the quarantine around Eientei." The Rabbit of the Earth lead the pair by the ready Reisen. "We have enough sick rabbits in the mansion as is, and we keep them confined to a makeshift quarantine wing. Crude, but effective, as long as everyone keeps protocol."

_46. The Aliens Speak_

Tewi slid open the rice-paper door and poked her head in. "Master Yagokoro, I've brought the vampire and her human maid," she called within. Remilia heard the occupant's indistinct words to her. "Oh. Yes, I'm still a knight. Thanks for inquiring about my health. Remilia is a knight, too." Tewi turned to Remilia and Sakuya. "The princess and pharmacist will see you now," she said, bowing slightly.

Inside was one Kaguya Houraisan, the Eternal Princess, sitting seiza with her long, beautiful robes arranged elegantly around her. Remilia half-expected track-pants and baggy tee, but she supposed Kaguya was still a princess after all. Besides her, sitting seiza in her red and blue livery, was one Eirin Yagokoro, the Brain of the Moon.

Remilia took her cushion and made a small bow towards the two. "Hello. Let's take care of the pleasantries right now. You already know I'm a knight. Might I enquire into your identities?"

Eirin spoke first, "The princess is a knave."

Kaguya followed, "Eirin is a knave."

Eirin spoke in her turn, "The princess is sick."

Kaguya responded in hers, "Eirin is healthy."

"Well, that just about covers it," Remilia said with a smirk.

Sakuya bowed. "Politeness demands that I greet you incourteously."

"Oh, did I mention Sakuya's a knave?" Remilia added.

_Solution: Eirin's statements are either both true or both false, so Kaguya is either a sick knave or a __healthy knight. In either case, she tells the truth, so both of Kaguya's statements are true. Eirin is thus a healthy knave, so both of her statements are lies, which makes Kaguya a healthy knight._

—/—

**Blaspheme Quarantine**

Kaguya took the implied insult with a titter of good humor. "A knave maid, huh?" she asked, still giggling.

"Yeah. Her deliberately teasing lies tend to get annoying."

Sakuya looked a bit embarrassed. "Sincere apologies for my behavior will be offered, m'lady."

"Which is, of course, false, because you cannot offer any sincere apologies even if you wanted to," countered Remilia.

Kaguya giggled again in sympathy before moving on to more important matters. The princess briefly turned to Eirin. "I will be handling the bulk of the explanations for now, Eirin," she informed. Eirin nodded, as she realized informing around her knavish nature would get rather tedious.

Kaguya returned her attention to Remilia and Sakuya. "As you may have heard, we have a wing set up for quarantine. Outside quarantine while inside Eientei, no rabbit should be sick. If we have reason to believe a rabbit is sick, then she is shuffled into the quarantine ward for isolation," began the moon princess in exile, "Patient rabbits, who should be sick, aren't allowed to leave the quarantine wing. They are attended to and monitored by nurses, who should be healthy. Both patients and nurses wear masks and gowns to minimize communicability both ways, though it sometimes makes it confusing which is which."

"I see."

"The wing has two entrances, segregated by type: one for knights' use, the other for knaves' use. Rabbits are notorious for not keeping detailed records, so it becomes quite a pain to make sure they haven't started lying or telling the truth when they went in doing the opposite. Segregation makes things much simpler."

Remilia felt the need to ask a few questions at this point, "How do you make sure a knave doesn't go through the Knight Entrance and vice versa? And how are you sure they aren't infected once they leave?"

"The two orderlies at the entrance ask them about their health status before they can enter, and ask them about their status and type when they leave," answered Kaguya.

"How does that help?" Remilia asked impatiently.

"You're a smart girl, Remilia. You'll figure it out," teased Kaguya with a good-natured smile on her face.

Remilia grumbled irritably for a minute before asking another question, "What happens if one of the nurses takes ill? Or if one of the patients turns out to be well? Or doesn't that happen?"

Kaguya became serious at this. "It shouldn't, but you never know," she cooly informed the vampire. "We call such rabbits _irregulars._ If if we find an irregular —that is, healthy— patient, then we take steps to have them released. They must be continued to be quarantined for a short time, though, to make sure they aren't carrying any latent infection, but once that expires, we release them. If a nurse takes ill, thus becoming an irregular nurse, then she is removed as a nurse, and she is returned to the wing as a patient."

The alarm bells at the back of Remilia's mind started screaming warnings. "Kaguya, I have this sinking feeling that you're about to ask me to ferret out these 'irregulars.' Is it true?" she asked, anticipating the answer.

Kaguya gave her a mischievous smile. "Your feeling is spot on, Lady Remilia! I _was_ about to ask you, but it seems redundant at this point."

"Damn it."

Kaguya shrugged. "I'm sorry, but Eirin won't allow me to do it myself, and we need this done both for the patients' protection, should they actually be well, and to keep Eirin's samples from being contaminated."

Eirin chose this moment to interrupt. "It will not slow down development for a cure to have contaminated data." she said, and Remilia automatically deciphered the knavese. "Also, the ward administered by Head Nurses Taaru and Hane is not most suspicious. Please investigate that ward in particular."

"I get your meaning, Eirin," observed Remilia. "Very well. We shall do this."

—/—

_47. Which Entrance?_

"Is this a quarantine wing or a bathhouse?" asked Remilia rhetorically as she stood in front of the so-called Quarantine Wing, which looked more like the front entrance to a Japanese _onsen_ than a place of healing like a hospital.

"It was most wise not to ask which entrance is for knights only and which one is for knaves only," Sakuya pointed out.

"I get it, I get it."

The two noticed a rabbit, presumably a nurse from her mask pulled down under her chin and writing something on a clipboarded chart. Remilia caught her attention, pointed toward one entrance and asked a single question, to which the nurse answered, "Yes."

Remilia nodded toward the place she had been pointing, saying, "Then that's the Knight's Entrance. That's where I must go. Sakuya, you take the Knave's Entrance."

"No, m'lady," Sakuya 'acknowledged' and the two stepped through their respective entrances.

The unidentified nurse had listened to this exchange and noted Remilia and Sakuya had made the not-completely-justified assumption that she was healthy. As it happened, she _was_ healthy, and so she shrugged to herself and continued her duties.

_Solution: Remilia asked, "Is this entrance yours?" If the nurse were a healthy knight, then she is to go through the Knight Entrance and she would answer truthfully, so the indicated door is the Knight Entrance. If the nurse were a healthy knave, then she is to go through the Knave Entrance and would answer with a lie, so the indicated door is not the Knave Entrance; it is again the Knight Entrance. As it happened, the nurse was not sick, so she is either a healthy knight or a healthy knave, and we do not have to consider the cases where she is sick._

_48. The Wing Guardians_

Remilia's feeling that the Quarantine Wing was just a converted bathhouse became greater as they saw the usual trappings of a bathhouse, with a changing room (allegedly for chancing into and out of protective gowns), refreshment dispensers, and the tall cashier's dais where two rabbit orderlies sat, complete with a bored look someone in their position in a bathhouse also usually had.

"Are you sick?" asked the two asked, almost as one, and the bored look never breaking once.

Remilia found herself answering, "No," just as she heard her servant answer, "Yes," to the same question. It appeared to satisfy the orderlies, as the waved them through.

It occurred to Remilia as she proceeded through this checkpoint that the vampire and maid pair didn't ascertain that nurse's health before rushing through. "Hey, are we using the right entrances?" she asked of her orderly.

Remilia's orderly actually cracked open her eye and made the following claim: "I am a knave, and at least one rabbit who has passed here claimed to be sick."

Sakuya's orderly made the complementary claim: "Either I am a knave, or every rabbit who has passed here claimed to be sick."

Remilia and Sakuya asked their gatekeepers almost simultaneously, "Are you a knight?"

"Yes," the two answered together.

Remilia's orderly held out a hand and added, "It'll cost you twenty five yen for the both of ya."

Remilia narrowed her eyes. "That's a lie, and a poor joke," she said, as the two proceeded confidently forwards, unworried now that they had chosen wrongly.

_Solution: Both rabbits are healthy. For a sick knight could not truthfully claim to be a knight, and a sick knave could not falsely claim to be a knight. Also, by their own answers, Remilia and Sakuya have proven themselves to be a knight and knave respectively. For a healthy knave cannot truthfully deny being sick and a sick knave cannot falsely deny being sick, and a healthy knight cannot falsely claim to be sick and a sick knight cannot truthfully claim to be sick._

_Remilia's orderly cannot be a knight because then her statement is a lie and she is healthy; she cannot lie if she's a knight, and therefore she is a healthy knave and here statement is false. Since she is a knave, then it must not be true that at least one rabbit who passed her claimed to be sick - all of them claimed to be healthy, and therefore knights. Thus, Remilia's entrance was indeed the Knight's Entrance. Also, since her orderly is a knave, it will not cost Remilia 25¥ to pass._

_Conversely, Sakuya's orderly cannot be a knave because that would make her statement true and she is healthy; she must be a healthy knight. Therefore, every rabbit who passed her in fact has claimed to be sick, and therefore knaves. Thus, Sakuya's entrance is indeed the Knave's Entrance._

—/—

**Unitary Blaspheme Quarantine**

Remilia quickly donned a gown and scribed a nametag for herself. As she was successfully able to write 'Remilia,' she surmised that either she was right-handed, or the special writing rules now hanging over Gensokyo didn't affect written fragments. Probably the latter, given that sentence fragments didn't affect utterances either. She shrugged and slipped the mask on before opening the door to the afflicted wing.

Sakuya stepped out of an adjacent door a moment later, bedecked in white gown and a nametag saying 'Sakuya.'

Remilia looked at Sakuya.

Sakuya looked at Remilia.

"What was the point of separating knights and knaves if they just throw them all into the same wing?" asked Remilia.

"I have at least one good idea, m'lady," Sakuya confessed.

"_*fu*_ Crazy lunatics," Remilia said, clucking her tongue.

"That said, overall I have no ideas," added Sakuya.

"In truthful words, no good ideas why the lunatics did this, but at least one idea — hence a bad idea."

"Incorrect."

"I don't want to hear this bad idea, Sakuya."

"Not understood."

—/—

**The First Ward**

Remilia and Sakuya entered the first ward, which contained about twenty rabbits all mingling about in nearly identical getups. As she feared, Remilia wasn't able to tell who the nurses were and who the patients were. "Crazy lunatics," she reiterated under her breath.

_49. A Pair of Rabbits_

Soon after entering, they were approached by several rabbits. The rabbits were identically dressed, and there wasn't any visual markers to indicate who was a nurse and who was a patient. Perhaps it was a secret marking that she couldn't sense. And if she couldn't sense it, Sakuya certainly couldn't sense it either.

Remilia noted one had a nametag of 'Kairi,' and asked, "Is Ami a patient?" She nodded to another rabbit with nametag 'Ami' that she noticed a bit before.

"No! Ami is a nurse!" Kairi snapped irritably.

Remilia ignored the rabbit's ire and strode towards Ami. "Is Kairi a nurse or patient?" she asked of the rabbit.

"Kairi is a patient," came her reply.

Remilia drummed her fingers against her upper arm for a few seconds in thought, then asked, "What would you say about Kairi's type?"

"I would say that Kairi is the same type as myself," Ami answered.

Remilia frowned. Remilia had found an irregular.

_Solution: Ami's second answer reveals that Ami and Kairi are indeed of the same type: both knights or both knaves. Suppose Ami were a healthy knight or a sick knave. Then she speaks the truth. Therefore, her claim that she would say that she and Kairi were the same type is true, and as such she would say she and Kairi were the same type and it would be the truth. Therefore, she and Kairi are the same type. Suppose instead that Ami were a sick knight or a healthy knave. Then she lies. Therefore, her claim that she would say that she and Kairi were the same type is a lie; she would instead claim that they were not the same type, and that statement would be a lie as well. Therefore, again she and Kairi are the same type._

_Suppose Ami and Kairi are both knights. Then if Ami is healthy, then Kairi is indeed a patient. If Kairi is healthy, then she is a healthy patient and thus irregular. If Kairi is sick, then her statement that Ami is a nurse is a lie - Ami is a patient who is healthy, and thus irregular. On the other hand, if Ami is sick, then her statement is a lie and Kairi is actually a nurse. If Kairi is sick, then she is a sick nurse and thus irregular. If Kairi is healthy, then Ami really is a nurse, and she is sick and irregular._

_Suppose Ami and Kairi are both knaves. If Ami is healthy, then her claim that Kairi is a patient is false and she is a nurse. If Kairi is a sick nurse, she is irregular and needs to be removed. If Kairi is a healthy nurse, then her claim that Ami is a nurse is false; she is a patient and healthy and thus irregular. On the other hand, if Ami is sick, then her claim that Kairi is a patient is true. If Kairi is a healthy patient, then she is irregular. If Kairi is a sick patient, then her claim that Ami is a nurse is true, which would make Ami a sick nurse and thus irregular._

_In all cases, at least one of Ami and Kairi are irregular, though we cannot tell who._

—/—

**The Second Ward**

Remilia and Sakuya dealt with that first pair and found nothing else of interest in the first ward, so they moved to the second ward next door.

_50. Another Pair of Rabbits_

As soon as they entered, two rabbits walked up Remilia and Sakuya.

The one with the nametag 'Usachi' declared, "I am a patient!"

"So am I!" declared a rabbit named 'Chiusa.'

"Yep, she's also a patient," agreed Usachi, pointing to Chiusa.

Remilia held up her hand. "Wait. Is either of you really sick?"

Chiusa said, "At least one of us would claim to be sick."

Usachi added, "At most one of us would claim to be sick."

Finally, Chiusa claimed, "I'm the healthy one."

"That's enough. I know one of you is exactly where she should be, but the other is irregular and must be dealt with," Remilia said.

_Solution: Usachi's agreement with Chiusa in her second statement tells us that they are either both lying or both telling the truth. Suppose they're both liars. Then by Chiusa's second statement, neither of them would claim to be sick, so they would claim to be healthy. This makes them both knights, as only knights can claim to be healthy. But then Usachi's third statement would imply that both of them would claim to be sick, so they're both knaves. So if they're both lying, then they are both knights and both knaves — a contradiction. So they can't both be lying and consequently must both be telling the truth, and exactly one would claim to be sick and is a knave._

_Because Chiusa claims to be the healthy one, she is a knight. Also, her statement is true, so she is a healthy knight, and she really is a patient and is irregular. This makes Usachi a knave, who is telling the truth and is therefore both sick and a patient, so she is regular._

_51. Hit and Run Claim_

After dealing with Chiusa, another rabbit popped up and shouted, "I am a patient!" She returned to sulking in the corner.

"What the…" Remilia murmured.

Usachi piped up. "Oh, that's Ruri! She's a knight."

Remilia knew immediately that Ruri needed to be dealt with, too.

_Solution: Usachi, a sick knave, claims Ruri is a knight, so she really is a knight. If Ruri is healthy, then she's a healthy knight and is therefore telling the truth, which means she's a healthy patient and irregular. If she's sick, then she's a sick knight and therefore lying, which means she's a sick nurse and irregular. Either way, she is irregular and needs to be dealt with._

_52. Four Rabbits_

Continuing within this Ward, the Scarlet Team came upon four rabbits, Umi, Kumi, Ari and Sari.

Umi said, "Kumi is sick."

Ari added, "Sari is also sick."

Kumi countered, "Umi is a nurse."

Sari finished with, "Ari is also a nurse."

Remilia didn't know what to make of this until she asked Usachi about it.

"Umi and Kumi are the same type, but Ari and Sari are different types!" she said.

Remilia took steps to see one was dealt with.

_Solution: In the case of Umi and Kumi, who from Usachi's statement are the same type, suppose first that they are both knights. If Umi is healthy, she is a healthy knight and is telling the truth when she says Kumi is sick. Therefore, being a sick knight, Kumi is lying when she says that Umi is a nurse; Umi must be a healthy patient and thus irregular. If Umi is sick, she is a sick knight and is thus lying when she says Kumi is sick, so Kumi is a healthy knight and is telling the truth when she says that Umi is a nurse. Umi should then be removed on account that she has taken ill._

_Suppose both rabbits are knaves. If Umi is healthy, then she is a healthy knave and is lying when she says Kumi is sick, so Kumi is in fact a healthy knave. Kumi is therefore lying when she says Umi is a nurse; Umi is a patient, and thus irregular. If Umi is sick, she is a sick knave and is telling the truth when she says Kumi is sick. Thus, Kumi is another sick knave and is telling the truth when she says Umi is a nurse. Therefore, Umi is a sick nurse and should be removed._

_In the case of Ari and Sari, who from Usachi's statement are of different type, suppose that Ari is a knight and Sari is a knave. If Ari is healthy, she is a healthy knight and is telling the truth when she says Sari is sick. Therefore, Sari is a sick knave, and is telling the truth when she says Ari is a nurse. If Ari is sick, she is a sick knight and is therefore lying when she says Sari is sick; Sari is healthy. Therefore, Sari is a healthy knave and is lying when she says Ari is a nurse; Ari is a patient and is sick._

_Suppose instead that Ari is a knave and Sari is a knight. If Ari is healthy, then she is a healthy knave and is lying when she says Sari is sick; Sari is healthy. Sari is therefore a healthy knight and is telling the truth when she says Ari is a nurse, so Ari is a healthy nurse. If Ari is sick, then she is a sick knave and is therefore telling the truth when she says Sari is sick. Therefore, Sari is a sick knight and therefore lying when she says Ari is a nurse; Ari is a patient and is sick._

_In no case is there grounds for removal of Kumi, Ari, or Sari, but Umi should definitely be removed._

_53. Three Rabbits_

Remilia and Sakuya were approached by three rabbits with nametags 'Banni,' 'Rabbi' and 'Haari.'

Banni began with the statement, "Rabbi is sick."

Rabbi countered with, "Banni is a nurse."

Banni continued with, "Haari is also sick."

Finally, Haari stated, "Banni is a patient."

"If Banni matches type with either Rabbi or Haari, she matches type with Rabbi," Usachi chimed in again from some ways behind.

Their statements lead Remilia to steps to see that Banni was removed.

_Solution: Banni either tells the truth or lies. Therefore, Rabbi and Haari are either both sick or both healthy. Since they make contradictory claims, they cannot both be knights nor both knaves; they must be a knight and a knave, and they are both sick or both healthy. Banni, being also either a knight or a knave, matches type with exactly one of them. Because of Usachi's true statement (remember, she is a sick knave and tells the truth), this last fact implies that Banni matches type with Rabbi._

_We learned in the case of Umi and Kumi (problem 52) that if two rabbits are the same type, and rabbit A claimed rabbit B was sick, and rabbit B claimed rabbit A was a nurse, then rabbit A is irregular. Therefore, Banni is irregular and should be removed._

_54. The Remainder_

After having Banni dealt with, Remilia turned to the remaining two. "What of you, then?" she asked.

Rabbi stated, "Haari is sick, too."

Haari said, "Rabbi is also a patient."

Rabbi replied, "So is Haari."

This time Remilia had Rabbi removed, but knew that Haari was right where she should be.

_Solution: We already know Rabbi and Haari are of opposite type from the previous problem. Furthermore, we already know that they are either both sick or both healthy. Suppose Rabbi is a knight and Haari is a knave. If Rabbi is healthy, then so is Haari. But this makes Rabbi a healthy knight who falsely claims that Haari is sick: a contradiction. Therefore, Rabbi cannot be a healthy knight. If Rabbi __is sick, she is a sick knight. But then Haari is also sick and so Rabbi is a sick knight who has made a __true claim: another contradiction. Therefore, Rabbi cannot be a knight; she is a knave and Haari is a knight._

_If Rabbi is healthy, then she is a healthy knave. Furthermore, Haari is also healthy, so she's a healthy knight. Therefore, Haari is telling the truth when she says Rabbi is a patient, which makes her a healthy patient and irregular. If Rabbi is sick, then she is a sick knave. By the same token, Haari is also sick, so she is a sick knight. Therefore, Haari is lying when she says Rabbi is a patient; she must be a nurse, and a sick one at that. She is again irregular and should be removed._

_Finally, if Rabbi is a healthy patient, she is a healthy knave, so her statement that Haari is also a patient is a lie — Haari's a nurse, and a healthy one. If Rabbi is a sick nurse, she is a sick knave, so her statement that Haari is a patient is the truth — Haari's a patient, and a sick patient at that. In either case, Haari is regular and no action need be taken concerning her._

—/—

**The Ward of Nurse Taaru and Nurse Hane**

Like the previous ward, Remilia and Sakuya systematically cleared this ward of sick nurses and healthy patients, with the help of Usachi. At the end, Remilia had to acknowledge her help. "Usachi, you have been extremely helpful. What can you tell me about the ward administered by Nurse Taaru and Nurse Hane?" Remilia asked.

Usachi considered for several seconds. "In the Quarantine Wing, a rabbit is considered _odd_ if she claims to be a patient. Every rabbit in that ward is known to all in the ward to be odd or not, and is claimed by each in the ward as being odd or not. Also, someone in a ward is _curious_ if all knighted patients and knaved nurses in that ward would claim she is odd, and all knighted nurses and all knaved patients in that ward would claim she is not odd." Usachi paused, as she knew she dumped something rather severe onto Remilia.

Remilia scratched her head as she wrapped her head around that. "Go on."

"Every rabbit in that particular ward has a best friend who is of the same type as her. Furthermore, if a rabbit, call her Mi, claims another rabbit in that ward, call her Yuu, was curious, Mi's best friend claims that Yuu is a patient."

"Anything else?"

"I know there is at least one rabbit who is a healthy knight in that particular ward."

Remilia bowed. "Thank you. Again, you have been most helpful."

Minutes later, Remilia was confronting Nurse Taaru.

_55. Nurse Taaru and Nurse Hane_

"Nurse Taaru, are your nurses healthy?" Remilia began.

Nurse Taaru nodded her head. "All of our nurses are in fact healthy," she asserted calmly.

"What of your patients? Are they all sick?"

Nurse Taaru scratched her chin. "At least one of our patients is sick," she hazarded.

Remilia walked away from the conversation feeling very uneasy. "Sakuya, is it crazy to think claiming only one sick patient is unreasonably modest?" she asked of her servant.

"Yes. Totally bonkers."

Remilia sought out Nurse Hane and interviewed her. "Are all your patients sick?" she began.

Nurse Hane looked offended. "Of course all our patients are sick!" she said hotly, "What kind of ward do you think we're running here?"

"All right. What of your nurses?"

Nurse Hane looked thoughtful for a second. "At least one of them is healthy," she said at last.

Again, an unreasonably modest answer. Remilia also knew she needed to confirm something. "Is Nurse Taaru either a healthy knight or a sick knave?"

"She is indeed one of those two."

Remilia put it together in a flash. Eirin was 'right' in her supposition that there was something wrong with this ward. Remilia realized to her horror that, in this ward, all of the nurses were sick and all of the patients were healthy!

_Solution: To start off, there is one rabbit who is a healthy knight in the ward. Let that rabbit be called A and let her best friend be called B, who is a knight. Then, if A claims B is curious, then B claims that B is a patient, that is, B is odd. Since A is a healthy knight, her claim is true, and B really is curious. This gives us a key fact: If B is curious, then B is odd._

_B is either odd or she isn't. Suppose B is odd. Then she claims to be a patient. But B is a knight, and the only knights that can claim to be patients are sick nurses or healthy patients. In either case, there is an irregular in the ward. On the other hand, if B is not odd, then she is not curious either, so the condition of all knighted patients and knaved nurses in that ward claiming B is odd and the condition of all knighted nurses and all knaved patients in that ward claiming B is not odd cannot both be true. If some knighted patient P does not claim B is odd, then P claims B is not odd, therefore her claim is true and P is healthy. If some knaved patient Q does not claim B is not odd, then she claims Q is odd, thus her claim is false and Q is healthy. If some knighted nurse N does not claim B is not odd, then she claims B is odd, thus her claim is false and N is sick. If some knaved nurse M does not claim B is odd, then she claims B is not odd, thus her claim is true and M is sick. In any case, there is either a sick nurse or healthy patient —that is, an irregular— in the ward._

_Remilia knew this before going into the ward. Nurse Taaru and Nurse Hane are each either telling the truth or lying. They cannot both be telling the truth, because we have already proved that at least one nurse is sick or at least one patient is healthy, contrary to their respective claims that all of the nurses are healthy and all the patients are sick. If Nurse Taaru is lying, then it is wrong that Nurse Taaru is either a healthy knight or a sick knave, so Nurse Hane is also lying, and vice versa. This means they are both lying. And if they are both lying, then it is untrue that at least one of their patients is sick and it is untrue that at least one of their nurses is healthy. Therefore, all of the patients are healthy, and all of the nurses are sick!_

—/—

**Breech in Quarantine**

Remilia's discovery of the secret of the ward of Nurse Taaru and Nurse Hane caused quite the stir in the Eientei. It was, after all, a serious error in assignment. Not only had all the nurses been exposed enough to be sick, but none of the patients had the sickness they had been isolated for. Rabbits scrambled to move all the (healthy) patients to the release isolation rooms and remove all the (sick) nurses from duty and quarantine them. It was a fiasco any way you looked at it.

"Well… this is most convenient," noted Eirin, watching the bedlam.

"I am envious of your ability to use sarcasm, Eirin," remarked Kaguya, quietly agreeing with the sentiment.

"I have similar feelings about Sakuya," Remilia commiserated.

"I would much rather have sarcasm than be able to tell the truth," Sakuya lied.

"As would I," lied Eirin. "How did we get so many missorted rabbits?"

Remilia hypothesized, "Perhaps those rabbits, being so poor at bookkeeping, were so incredibly sloppy at keeping the quarantine intact that a lot of mix-ups occurred."

A few passing rabbits within earshot blushed.

_56. Eirin Takes Ill_

Kaguya's mind had drifted off to other things, however. Specifically, to Eirin. "Eirin, are you all right."

Indeed, Eirin Yagokoro was looking flushed and was suffering and ill. "Uh…" she groaned and stumbled to the ground.

"Eirin!" Kaguya shouted, almost leaping to Eirin's side and noted her flushed features. Kaguya held her palm to Eirin's forehead. "Eirin, you're running a fever!" she declared in shock.

Eirin made a few mumbling noises, then gathered her strength to make a statement. "After… saying… this… statement…" she struggled to say, "I… I'll know… whether or not… I've become sick."

"What are you talking about, Eirin?" came Kaguya's panicked question.

Remilia had just finished processing the implications of Eirin's statement. "Get away from her, Princess!" warned the vampire, "She has indeed become sick!"

_Solution: Eirin's statement —which we shall label S— is, "After saying this statement, I will know whether or not I've become sick." We will first prove that Eirin knows the truth of S as she states it, as per requirement for her as a sick or healthy knave._

_We note that S is either true or false. Either Eirin will know her health status after stating S, or she will remain in the dark about her health. Ignoring now the exact contents of S, if Eirin states S and notes it to be false, then as a knave she knows that she is still healthy. On the other hand, if she states S and notes that it is true, then as a knave she knows she has become sick. In either case, Eirin knows whether or not she is sick after making statement S. But this is exactly what S asserts: that Eirin will know after making S whether or not she has taken ill! Therefore, Eirin knows at the time she states it that S is indeed true, and can therefore state either it or its negation at the time she makes her statement._

_Since has stated S and not not-S, and S is true, as a knave Eirin cannot still be healthy; she must have become sick!_

_(The complementary sentence that proves Eirin is not sick is T, "After saying this statement, I will not know whether or not I've become sick." From the previous derivation, we know ahead of time that it is false — Eirin will indeed know whether or not she is sick after stating the sentence.)_

—/—

Kaguya obediently retreated far from the stricken Eirin, but she managed to bark, "Get Eirin to an infirmary bed! Get up an individual quarantine around her!" Rabbit nurses, bundled in their gear, rushed in and gathered up the ill Eirin and carried her out of the room. She then went off to parts unknown. Remilia and Sakuya followed the rabbits carrying Eirin.

About ten minutes after Eirin was tucked into one of the infirmary's bed, Kaguya returned bedecked in the same protective gear as the rabbit nurses. She carried a pail of ice water and a stack of towels. After seating herself, she soaked a towel in the ice water and laid it across Eirin's forehead.

"I guess even your NEETery knows some limit, Teruyo," observed Remilia a bit snarkily.

"Shut up! And don't call me 'Teruyo!'" barked the princess. Then, remembering herself, she sighed and said, "We need your help to figure out who broke quarantine."

_57. Myxomatotic Rabbits_

Kaguya pulled some pages out of a manilla folder. "This is what we found out so far," Kaguya said, handing Remilia a stapled stack of papers. "Four rabbits, whose names have been withheld and designated A, B, C and D, may have exposed Eirin around six hours ago, the incubation time of the disease. Now, at least one of these rabbits was sick and showing symptoms at the time of Eirin's exposure. These are their depositions."

Remilia took the ream and started skimming through it. "Are all these rabbits showing symptoms now?" she asked.

"Yes. That's part of why we were able to tell that one of them broke quarantine around that time."

Remilia tapped her chin in thought for a minute. "That eliminates the dual effect of sickness and health by reversing the roles of knights and knaves," she reasoned aloud.

"I see how that would help." Kaguya sobered up even more. "I would like to know who is responsible for this mess. In particular, I'd like to know if D was one of the ones responsible for breaking quarantine. She's a notoriously sloppy individual and regularly puts others in danger. If she's in any way responsible, I'll lock her up until this epidemic is over."

Remilia nodded, and read through the depositions:

* * *

_Officer Houraisan: A, were knights involved in breaking quarantine?_

_Rabbit A: At least one knight was involved._

_Officer Houraisan: B, are both you and A knights?_

_Rabbit B: Yes._

_Officer Houraisan: A, are you the same type as B or C?_

_Rabbit A: No._

_Officer Houraisan: D, how many knights are there in your group?_

_Rabbit D: There are no fewer than two knights amongst us._

_Officer Houraisan: D, could C have breeched quarantine without your help?_

_Rabbit D: No._

_Officer Houraisan: C, could B have breeched quarantine without your help?_

_Rabbit C: Yes._

_Officer Houraisan: A, are you in any way responsible for the breech of quarantine?_

_Rabbit A: No._

_Officer Houraisan: B, are you in any way responsible for the breech of quarantine?_

_Rabbit B: Only if I am a knight!_

* * *

"'Officer Houraisan'?" Remilia asked, questioningly.

"I am the highest royalty here. I can give myself any title I wish."

"Feh," tsked Remilia. After studying the deposition in ernest, she came up with a verdict. "You are in luck, lunatic princess. I know two are responsible for breaking quarantine and exposing Eirin, and I'll have you know that rabbit D is indeed very responsible."

_Solution: From her first answer, B cannot be a sick knave, because a sick knave could never claim to be a knight, so she must be a sick knight. But B claims that A and B are both sick knights, so if A were also a sick knight, this would give us a truthful sick knight, so A must be a sick knave. If either of B or C were of the same type as A, then A would answer yes to Kaguya's second question to her; since she didn't neither of B or C are knaves. They must be sick knights. (We already knew B was a sick knight.) Finally, D claims there are no fewer than two knights amongst A, B, C and D. There are indeed at least two knights, so D's statement is true. Therefore, she is a sick knave._

_A is telling the truth when she says a knight was involved in breaking quarantine, and she was not responsible for it. This means someone other than her is responsible for the breech. Since the only other knights involved are B and C, at least one of them is responsible for the breech. D is telling the truth when she says that C couldn't breech quarantine without her help, and C is lying when she says that B could have breeched quarantine without hers. Therefore, if B breeched quarantine, C was involved, and if C was involved breeching quarantine, D was also involved. Because at least one of B and C are responsible, either possibility ends up with D involved in breeching quarantine._

_Since B claims that she is responsible only if she is a knight, her claim is false as a sick knight. However, she is indeed a knight, therefore she is not responsible. Therefore, C and D alone are responsible for the breech._

—/—

**The Terror of the Vorpal Bunny**

_"KIIII-YAAAHHHHHHHH!"_

A scream tore through the air, followed by a heavy thump-thumping of a pair of panic-propelled feet thundering through the mansion. Into the room crashed Tewi, who looked absolutely wide-eyed and terrified by… something. The rabbit dived toward Kaguya, latching onto her like a barnacle. "No! Nonono! Keep it away! Keep it away! It… she … _*hic*_ … wants to kill me!"

"Tewi, what's wrong?"

"She came out… transformed… then… then…" She returned to crying into Kaguya's dress. "…came after me!"

Kaguya silently parroted her words, realizing slowly what happened to the unfortunate luck-giving rabbit. "Tewi! Who attacked you!" she demanded.

"A great silver beast… red eyes…" described Tewi haltingly. "Long ears like a rabbit… but… but… she had fangs like a wolf… slobbering for my flesh…" Her eyes were overflowing with her frightened tears, obviously difficult for her to describe the horror she had confronted just minutes ago. Her actual age no longer mattered — right now she was a child who was scared half to death.

"Sounds more like a beast than a person," murmured the Scarlet Devil.

"She said I was a mean-spirited trickster who deserved to be made into rabbit stew!"

Kaguya mused aloud. "Well, there is a grain of truth to that…"

_"Waaahhhhh!"_ Tewi completely broke down into tears.

"That sounds like a lagotheramorph…" Eirin rattled.

"Eirin? You're awake?" asked Eirin obviously.

The sick Brain of the Moon opened her bleary eyes to glare balefully at her. "Who could sleep through that racket?" she asked rhetorically, then took a moment to cough out her throat frog.

"What is a 'lagotheramorph'?" Remilia asked.

"A Vorpal Bunny," Eirin said. For some reason, the picture in Remilia's mind was one of a cute little bunny rabbit with a bloodstained maw ferociously chewing the heads off of armored knights. And nasty, pointy teeth. And for some tangential reason, coconuts were involved. Remilia didn't think it showed on her face, and if Eirin saw anything, she didn't comment on it, but continued, "One of the rabbits is suffering from lagotheramorphism, AKA, Vorpal Bunny Syndrome. I was afraid this might happen eventually, but I was hoping that this settlement's small size and a quick remedy might beat the odds."

She waved off further comment of that. "Never mind. There's no helping it now. I must continue my research or other rabbits will suffer the same fate." Eirin took another moment to cough up her lung. "Remilia, Sakuya, please find the lagotheramorph… like a werewolf, or maybe the werehataku, she will appear as normal until she reveals herself. However, she will be sick in the normal way while in that form." Eirin let out an exhausted breath. "That's all I have for now. Let me rest," she groaned then settled into a restful position.

—/—

Hours of careful investigation later revealed that there were thirty rabbits who might have been the Vorpal Bunny. They were all gathered behind a partition to separate them from a spot where Tewi would identify the culprit. Right now, 'Officer Houraisan' was having a grand old time herding rabbits about.

"The plan is simple! We have Tewi identify the Vorpal Bunny out of small groups of about ten or so rabbits," announced 'Officer Houraisan'. "Then we lock her up and wait for Eirin to create a cure!"

"You call that a plan?" Remilia asked, disbelieving. "That hardly qualifies!"

"If you want to interview every single one to see if they are knights, knaves, healthy or sick, as well as being the Vorpal Bunny, be my guest!" Kaguya said cheerfully, albeit with a touch of annoyance.

"No thanks."

"Good! Show in the first group!"

The first group was ushered into the lineup by Reisen. The door opened, and a jumpy looking Tewi was lead into the room. She took one look at the gathered rabbits and screamed her head off. She was quickly lead out of the room. The rabbits looked at each other quizzically.

"I think we best not use her as a Vorpal Bunny detector," concluded Kaguya.

"Agreed. Her screaming is getting on my nerves," Remilia affirmed.

"At least we narrowed the culprit down to these nine rabbits," pointed out Sakuya. The two knights murmured that it was actually ten rabbits, but with unspoken agreement with what she meant.

_58. Two Plus Reisen_

Remilia and Kaguya agreed to split up the work: each would interrogate five rabbits. After that, they would combine their chief suspects and interrogate them together.

"Show in Hitomi and Onari, Reisen," said Remilia, sitting down on her side of her room's interrogation table.

Hitomi said, "I am the Vorpal Bunny!"

Onari said, "I am the Vorpal Bunny!"

"Reisen?" prompted the vampire.

"If you include me, exactly one of us rabbits here is a healthy knight or a sick knave," said the lunar rabbit.

Remilia raised an eyebrow. "Can I be sure you haven't contracted the sickness?"

Reisen looked offended. "Come now, you heard me tell the truth that Tewi was healthy a little while ago."

Remilia nodded, accepting her answer. A beat later, she realized that she need not have taken Reisen's health for granted. Whether Reisen was well or ill, Remilia knew the Vorpal Bunny was neither Hitomi and Onari.

_Solution: Hitomi and Onari cannot be both telling the truth, so they cannot both be either healthy knights or sick knaves. If one is the Vorpal Bunny, then she is the one who is a healthy knight or a sick knave, and if she's the Vorpal Bunny, she's sick, so she must be a sick knave. If Reisen is healthy, then there are two rabbits in the bunch who are each either a healthy knight or a sick knave, which makes Reisen a liar; if Reisen is sick, then there is only one rabbit in the bunch who is either healthy knight or a sick knave, which means Reisen is telling the truth. Either case results in a contradiction, so neither Hitomi nor Onari is the Vorpal Bunny._

_59. Three Rabbits in a Stew_

After dealing with the first two rabbits, it was time to tackle the rest of her lot: three rabbits, Arisa, Satako, and Chidori.

Arisa said, "If any of us three is the Vorpal Bunny, it's Chidori!"

Satoko said, "I am not the Vorpal Bunny."

Chidori said, "At least two of us are each either a sick knight or a healthy knave."

Remilia thought a bit, then said, "Obviously, you are not the Vorpal Bunny. You are dismissed." That rabbit bowed and took her leave. It was time for the second round, and the two remaining rabbits spoke in the order that they spoke in last round.

The first stated, pointing towards her fellow, "She is a knave. She is also healthy."

The second pointed towards the first. "She is healthy," she said.

Remilia thought about this a while, but found she had no reason to detain any of them. "I hope Kaguya is having more luck sniffing out the Vorpal Bunny than I am," she grumbled.

_Solution: First, we must find out who Remilia dismissed. Suppose Chidori is either a healthy knight or a sick knave. Then it is true that Arisa and Satoko are each either a sick knight or a healthy knave. Thus, Satoko is lying when she says she is not the Vorpal Bunny, and therefore is the Vorpal Bunny. Suppose instead that Chidori is either a sick knight or a healthy knave; then there are at most one of either a sick knight or a healthy knave in the group: Chidori herself. Then Arisa is telling the truth. If we suppose the Vorpal Bunny is indeed in this group, it must be Chidori. In either case, Arisa is not the Vorpal Bunny. This is the rabbit Remilia dismissed._

_We've seen above that if Chidori is a heathy knight or a sick knave, then Satoko is a sick knight or a healthy knave, and if Chidori is a sick knight or a healthy knave, then Satoko is a healthy knight or a sick knave. Either the two match in exactly one of their healthiness or their type; if they match both, or don't match in either, they either both lie or both tell the truth, and this is not the case._

_In the absence of Arisa in the second round, Satoko spoke next, followed by Chidori. Satako claims separately that Chidori is a knave and Chidori is healthy. If she's truthful, then Chidori really is a healthy knave; if not, then Chidori is a sick knight. Either case means Chidori is a liar and Satoko is actually sick. If Chidori is sick, she's a sick knight, which means Satoko is a sick knave. But sick knaves tell the truth, so Chidori cannot be a sick knight. She must be a healthy knave. Since the two do not match in health, they must match in type: Satoko is a sick knave._

_Since Chidori is healthy, she isn't the Vorpal Bunny. And since Satoko tells the truth, she isn't the Vorpal Bunny either by her first statement._

—/—

Kaguya called a break to the interrogations to check on Eirin. Remilia didn't have anything to do, having eliminated all of her group, so she accompanied Kaguya, all the while trying to pick her brain to see how far along she was.

To which Kaguya had answered nothing more substantial than, "I'm making progress."

This was immensely unsatisfying to Remilia, who wanted to know how many rabbits she had eliminated from her pool. The vampire was beginning to think she hadn't made much progress at all, although it was true that she had made some progress.

"Eirin, how are you doing?"

Eirin stirred from her bed. She was still looking a bit flushed and she had acquired a mild bed-head. "Better. Well enough to work, but I still feel crappy." Her voice was raspy and grating, but it sounded much better than a couple of hours ago. "I've thought it over, and I've realized that the Vorpal Bunny will be a knight."

"Why do you say that?" Remilia asked.

"The lagotheramorph stated an almost-truth, but 'almost' —as they say— only counts in horseshoes and _kyuun's_. As the lagotheramorphism progresses, the effects of her sickness will diminish, but her perceived certainty will increase as slight delusions set in. She'll start talking less like a knight and more like a normal, if honest person." Remilia guessed that something showed up on her face that time, because Eirin addressed her unspoken question, "And before you ask what a _'kyuun'_ is, let's say your sister Flandre is rather infamous in her own right, Lady Scarlet."

Eirin maneuvered herself stiffly out of bed, despite the princess's objections. "Eirin, you must rest! Your body is fighting off an infection!" she pleaded.

Eirin's reply was stern, even through her raspy voice. "No, I must _work._ If we have a Vorpal Bunny about, then we need a cure desperately. I can't afford to be lying down now. Eientei can't afford it either." She smiled, despite her obvious infirmity. "Fortunately, I worked almost all of it out while I was sleeping. I should have a treatment for this plague and our lagotheramorph within the hour."

"Thank goodness for that!" the princess sighed in relief.

"Princess, you should worry more about _finding_ the lagotheramorph than on me," the researcher said, letting a bit of impatience creep into her voice.

"Quite so. We're no good to Eirin here," agreed Remilia, "So let's call the break over and I'll help you weed through your remaining rabbits."

"Okay!" chirped Kaguya cheerfully.

—/—

"So, in the end, you were able to eliminate exactly none of them," said Remilia as she looked at the completely undecreased pool of rabbits that the moon princess was (un)able to pare down. Kaguya just laughed bashfully, scratching her head in embarrassment. "Were you at least able to determine if there were any knaves amongst them and how many?"

"It was either two or three," Kaguya answered.

"I suppose that _is_ progress, even if it is only slight," Remilia sighed with annoyance. "And who were they?"

"I don't remember! _*teehee*_"

"Don't 'tee hee' me!" Remilia growled. "What about sick rabbits?"

"The same range, two or three."

"And who were they?" Remilia asked flatly, expecting a lame answer.

"Again, don't remember!" came Kaguya's wholly expected answer.

Remilia boiled over in frustration. "You're… Sakuya!"

"You're a real help!" Sakuya said, supplying the sarcasm.

"Sorrieee!" drawled the lunatic princess.

"I'll handle them. You… try not to swallow your tongue or something!" Remilia said irritably, shooing her away.

"That's mean!" Kaguya pouted. The vampire brat ignored her. Seeing that she would get no further with Remilia, the moon princess turned to Reisen. "Reisen, go check on Tewi and make sure she's all right." Reisen bowed and left without a word. "How rude! Well, I'll get Eirin to shove a huge suppository up her butt later. Anyway, I'm gonna go back to seeing to Eirin."

"That was much more than I wanted to know about how you keep discipline in this place," mumbled Remilia.

_60. The Princess's Set_

"All right you five, let's get started," Remilia began, "Make a statement germane to finding the Vorpal Bunny."

"I am not a knave," said Yuki.

"I am a knave," countered Yui.

"Yuki and Nene are both knights," said Mimi.

"If Yui is a knight, then she is the Vorpal Bunny," added Nene.

"I am not a sick knave," concluded Rina.

Remilia pointed to one of them and said, "You obviously aren't the Vorpal Bunny, so you are dismissed."

The dismissed rabbit did not immediately leave but approached Remilia. She whispered into the vampire's ear, "If neither Nene nor Yui is the Vorpal Bunny, then neither is Mimi." Remilia nodded her acknowledgement, and the dismissed rabbit skittered away satisfied.

The rabbits made statements in the same order as last time.

The first rabbit said, "I am sick. Mimi isn't, though."

The second rabbit added, "Nene is sick."

The third said, "There were exactly two knights in this group."

The forth claimed, "I am sick."

Remilia's eyes narrowed in thought as she realized the Vorpal Bunny had escaped her clutches.

_Solution: Rina is a healthy knight. For if she were a sick knave, then she would be lying and sick knaves do not lie, and if she were a sick knight or a healthy knave, she would be telling the truth, and both only lie. Since the Vorpal Bunny is a sick knight, Rina cannot be the Vorpal Bunny. This is the one Remilia dismissed._

_There is exactly one Vorpal Bunny. So if either Nene or Yui is the Vorpal Bunny, Mimi cannot be. But Rina's statement means that if neither Nene nor Yui is the Vorpal Bunny, then Mimi isn't either. Therefore, Mimi is not the Vorpal Bunny._

_The first rabbit in the second round of statements is Yuki, followed by Yui, Mimi and Nene. We already know Yuki is healthy, as she denies being a knave; so it is not the case that she is sick as she claims. Therefore, she is a healthy knave and is lying, thus her claim about Mimi not being sick is also false — she is indeed sick. It also means Mimi is lying when she claimed Yuki and Nene are both knights in the first round of statements. Therefore, Mimi is a sick knight. Because of this, we also know she is lying when she claims there were exactly two knights in the original group (that is, Rina and Mimi); there is a third knight we haven't yet found._

_Nene cannot be this knight, because she claims to be sick and is thus a knave. Therefore Yui's the knight, and because she claims she is a knave, she is also sick. Yui is lying claiming that Nene is sick; Nene is healthy, which makes her a healthy knave and her conditional statement is false. Therefore, Yui is not the Vorpal Bunny. Furthermore, because Nene and Yuki are healthy knaves, neither is the Vorpal Bunny. We have excluded all of the possible rabbits who may be the Vorpal Bunny, therefore, the Vorpal Bunny is not in this group either!_

—/—

"What the hell!" Remilia exploded, "The Vorpal Bunny isn't in this group either!" Remilia began pacing about the room, ranting her head off. "Yet, it is _this_ batch that Tewi went bonkers over, so she _must_ have seen the rabbit who was the Vorpal Bunny!"

Sakuya was silent as Remilia raved at her inability to discover where the lagotheramorph was. She was thinking. Thinking about the hidden subtlety in the situation and the flaw in Remilia's argument that they must have had the Vorpal Bunny amongst those they interviewed. "That time, Tewi saw no more than ten rabbits in that room," she announced at last.

"Reisen? She can only be the lagotheramorph if she's sick, and I'm _quite_ sure she isn't," Remilia scoffed.

"Are you _sure,_ M'lady?"

"Positive."

Sakuya stared at Remilia as if she was sure her mistress was forgetting something. Finally, the knave said, "Reisen said that time that Tewi was still healthy and may pass."

Remilia almost agreed with the elegant and knavish maid before the realization struck the young vampire, her eyes shooting wide open. "I've been a fool!" she shouted, and took flight towards the rabbit residences.

—/—

Kaguya blinked in surprise. "You mean you _didn't_ tell Reisen you were expecting Remilia and her maid?" she asked.

"No, because I was then a healthy knave…" She paused to clear her throat. "…and I _was_ indeed expecting them!"

"You didn't drop any knave-hints, either?" Kaguya probed.

"No, I never talked about it! There was no way…" Eirin took a moment to cough. "…Reisen could have known I was expecting them." Eirin was silent for a few moments as that worked through her illness-addled brain. "But that means…"

"Oh no! Reisen is sick!" Kaguya declared in realization.

Eirin's face was grim as a more disturbing possibility opened up before her. "Worse than that! Remember that Tewi was terrified when you showed her that group of rabbits? Reisen was amongst them!" she pointed out.

Kaguya put it all together at that moment. "Then… Reisen is…"

—/—

"You escaped me before, Tewi, but you will not do so again."

Tewi cowered in the corner, shaking like a leaf caught in a hurricane. In the doorway was Reisen, her eyes burning a beastly amber instead of its normal red color. Tewi opened her mouth in a silent scream as Reisen began her transformation.

White fur sprouted over Reisen's body, covering her in a silver coat instantly. Her physique grew, straining her uniform with increased musculature and frame. Her head altered, resembling more a rabbit's head, but no rabbit had a heavy jaw full of sharp teeth. Her hands turned into paws sprouting sharp claws.

The entire package promised pain and death. Tewi believed that promise, and whimpered in fear.

"Don't cower, you cur. Take your punishment like a rabbit!" hissed the Reisen-monster, holding up a huge, clawed paw ready.

_"STOP!"_

A flash of movement smacked into the monstrous Reisen's jaw, knocking her backward.

The creature growled as she found Remilia, Sakuya and Kaguya interposed between herself and her prey. "Stand aside, vampire! I will not be denied!" hissed the beast-rabbit.

"You'll have to go through us first, Reisen Udongein Inaba," sneered the vampire confidently, trash talking as much as her knighthood allowed.

"Then I will!" snarled the half-rabid beast, and bolted for Remilia.

The vampire realized immediately that this creature was not something to be taken lightly. Reisen the Vorpal Bunny was much faster, stronger, and hardier than Reisen Udongein Inaba. Its fangs nearly got a piece of her as she reacted on instinct to the Vorpal Bunny's lunge. All the Vorpal Bunny got was a mouthful of pink cloth.

"M'lady!" Sakuya whipped out a spell card and declared it.

_**"~Illusion World [ The World ]~"**_

In Sakuya's Frozen World, innumerable knives appeared around the altered Reisen. _"Toki wo ugoki de arimasen!_ (Time shall not move!)" lied Sakuya, and time started to move in accordance to her flipped command.

With time moving, the world around Sakuya began moving again.

Including the knives.

Including the beast's mouth.

Which smiled.

The knives bolted from where they were frozen and through the rabbit-beast. Passed through as if either they or the beast were illusion, not as if the knives were cutting flesh to clear a paths for themselves.

"It's possible!" gasped Sakuya.

"It certainly is, especially if you are not aiming where you think you're aiming," chuckled the monster before it blurred suddenly. Just as suddenly, Sakuya found herself smacked smartly and wheeling back out of control and into a wall, where consciousness left her.

"Sakuya!" blurted Remilia, then turned her furious gaze at the Vorpal Bunny. "Damn your lunatic eyes, Reisen!" Remilia instantly deciphered what had happened. The monster used Reisen's ability to mesmerize others with her lunacy-inducing eyes to distort Sakuya's senses.

"I can use my illusions to cloak myself, hiding me as easily as the kappa hides within her camouflage," chuckled the Vorpal Bunny evilly as she faded from view. Remilia stood in a wide, ready stance, straining her senses to try to find an imperfection in Reisen's cloaking illusion.

A slight artifact in her illusion revealed the lagotheramorph's position. It wasn't enough to let Remilia dodge, but the vampire blocked the blow, which knocked her back without doing serious harm to her. The knock back gave the Scarlet Devil enough play to deploy her own spell card, and the artifact she spotted allowed her to slice clear through the Truly Mad Moon Rabbit's illusion.

_**"~Divine Lance [ Spear the Gungir! ]~"**_

Remilia's aim, unlike Sakuya's, was as true as her statements. Reisen's illusion was broken around her. However, Remilia's strike cost her reflex time — time enough for Reisen to close the gap with her enhanced speed and land a powerful strike to the vampire's neck.

Careening from the impact, Remilia slammed bodily into a sturdy support beam and did not go through rice-paper walls, letting out a grunt at the impact. She slumped as she landed on the floor, the wind knocked out of her.

The lagotheramorph chortled darkly, and turned to find… Kaguya Houraisan interposed between her and Tewi, looking defiant and badass as she pointed her bejeweled Hourai branch at the monster. "Ultimately, it's _my_ responsibility to defend those under me!" she announced loudly and sternly.

They were big words. Unfortunately, the lunatic princess was in no shape to make good on them, coming from a long series of dilettantes. The lagotheramorph was much faster than her, and of course, Reisen had some combat training. Kaguya was easily swatted aside.

"Remilia! Sakuya! Princess!" squealed Tewi, seeing her guards taken out one by one.

"That's done," rumbled the monster, "No one remains to defend you, you roguish rabbit."

"Reisen, if you go through with this, you'll regret it later."

"No, I don't think so," rasped the altered rabbit. "You've had this coming to you for a long time, you fibbing trickster!" The venom in her voice was as vile as the slobber she drooled. "You're an imp who wishes nothing less than to make my life miserable! There's nothing but malice in your heart for me, so don't find it surprising that it's reciprocated!"

"Please don't do this, Reisen! I don't hate you! I _never_ hated you!" pleaded the terrified trickster rabbit, "My pranks are just to get your attention… and you're so fun to pick on, but that's because I like you!"

The Reisen-monster was not fazed by Tewi's words. "You're a dirty, lying little plague rabbit, who's just saying that to save your skin!"

"Reisen, you idiot! I'm a _knight_!"

The Reisen-monster stopped as Tewi's loud declaration penetrated the red-tinged fog surrounding her brain. Tewi claimed to be a knight, and she really was a knight, and therefore she was healthy and telling only truths. Gensokyo may be a place not bound by common sense, but for the moment, it was bound by logic. That logic froze the creature's brain, as it tried to rationalize the delusion against the logic it knew to be true.

That moment pause was enough.

A frail-looking but very sturdy arm wrapped around the lagotheramorph's neck and slammed her powerfully down. "It is not my fate to be bested by a _mere rabbit,_ even a mutant one!" Remilia Scarlet growled. "Eirin, NOW!"

Suddenly, the monster's eyes went big and round with surprise. The lagotheramorphic Reisen looked behind her in alarm as Eirin stood at her rear, with a demented grin on her face and holding a huge suppository 'in position.' "Get away from her, you _**doe!**__"_ she growled, and rammed the payload home.

Reisen the Vorpal Bunny let out a high-pitched squeal of agony as the suppository entered her body, then fell over limp as both the pain and the medicine took their toll. The transformation process reversed itself, turning the great hairy beast back into the lunar rabbit they all knew.

Kaguya, who had been pale facing the monster, then horrified at the application of an oversized suppository, finally processed all that had happened in the past few seconds. "Eirin? Is that…?" Kaguya trailed off in hopeful anticipation.

Eirin smiled and said, "I am a knight!"

Kaguya threw up her hands in joy. "It is! Eirin, you're wonderful!" she squealed, hugging the Brain of the Moon tightly.

—/—

"Reisen used a particular manner of speech that hid the fact that she had become sick. By saying 'I would say _such-and-such_,' it implied that the such-and-such was true, although the actual sentence was false," Remilia explained to the princess. "Since the such-and-such was true, she could not, in her condition, say such a statement, which meant that her sentence would be a lie and she could say it in her condition. Once she 'established' herself as a truth-teller, she could freely refer back to that lie to forge her credentials."

The princess giggled. "I see. Very clever of Reisen. Though it disturbs me that she actually wanted Tewi dead."

"I suppose she wasn't in her right mind. Now that that business is concluded, time for us to leave," declared Remilia as she stepped off the verandah of the main room. Eirin had administered a prophylactic to her and Sakuya (as well as the princess) so that they could leave immediately without breeching quarantine. She paused, thinking for a moment, the turned toward the Mad Pharmacist, stating, "Oh, and next time you create a plague like this, I'll come to personally beat the snot out of you."

"I will forget your warning immediately."

"I'm glad to hear that lie," replied Remilia with a smirk. "Come Sakuya. Let's go."

"M'lady."

Remilia let out a noisy sigh as they arced high into the air. "I'm _beat_!" the vampire kid declared as they banked toward the Misty Lake, "Plague-ridden rabbits, crazy lunarians, plague-ridden crazy lunarians, and a plague-ridden crazy lunarian were-rabbit! It's enough to drive one to as much lunacy as _they_ exhibit!"

They flew in silent contemplation for a while. "I'll want a soothing tea when we get home. Anything interesting?" Remilia asked suddenly.

"No, m'lady. I have just obtained some high-quality tea from Kourindou last week, and it's a right nasty brew," informed the maid.

Remilia smiled a toothy grin. "Heh! Liar!"

—/—

**Interlude: Reisen in Recovery**

"So, how do you feel, Reisen?"

Reisen looked up to see Tewi standing in the doorway. She smiled at the white, black-haired rabbit. "Much better, thank you." Reisen paused, contemplating her own words. "I'm a… knight?" she said, tentatively.

"That statement proves you're healthy once more," said Tewi.

Reisen's finger went to her cheek as she thought about it. "Yeah, I guess it does," she concurred. Reisen cast her head down, looking thoughtful for a while, before starting to speak, "Tewi… about my behavior…"

"Don't worry about it. And don't apologize. I know it was just the effects of the plague," Tewi waved off Reisen's concerns. "I must say, this episode has made me re-think my pranks on you."

A warm feeling blossomed inside Reisen's breast. Was this finally the long awaited connection to her fellow rabbits? "Does that mean you'll let up on your pranks?" she asked eagerly.

Tewi's smile was anything but kind. "Oh, no. My next prank will be _epic!_"

The warm feeling became a chill down Reisen's spine. "But… I thought you agreed that it was just the plague's doing!" protested the moon rabbit.

"Oh, I did!" confirmed the roguish, if knightish rabbit. "That doesn't mean that the whole episode doesn't _piss me off._ Also, meaner pranks for a while may desensitize you to future harmless ones."

"Aw, come on! Surely this will be counterproductive! What if the Master hears about this?" protested Reisen, starting to shake.

"Oh, don't worry. She said, and I quote, 'Tewi, you do _not_ have my full permission and blessings to do this!'"

Reisen paled at these words, as Eirin's knavish nature was well-known to her.

"Now, where did I put Mr. Thingy?" Tewi wondered aloud, stepping casually out of the infirmary to seek that item. Reisen cried a high pitched call of terror.

_(To be continued… unless I have the plague!)_

* * *

Some chapters just gel and are easy to write. Other chapters just kick your ass. I won't lie. This chapter kicked my ass. I wasn't really satisfied with how it was wrapping up until the very end.

"Blaspheme Quarantine" is the tenth level of the original _Marathon_ game. Yes, the same game that Remilia and Sakuya commented about the beginning of the chapter.


	6. Remilia in the Forest of Insanity

_Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.

All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond Smullyan.

Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.

Remilia Scarlet, the Scarlet Devil, has awoken to find herself unable to lie, and therefore a knight. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.

It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.

Remilia and Sakuya have just recently helped contain an outbreak of a disease amongst the Eientei rabbits that caused knights to become liars and knaves to tell truth. This was complicated by Reisen Udongein Inaba contracting a more serious form of the disease, which turned her into a brutal monster, the Vorpal Bunny. The sub-incident was resolved by the timely creation of the cure by Eirin.

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 6:_

_In Which Remilia Scarlet Hikes Through the Forest of Insanity_

* * *

Remilia had been enjoying her mid-evening tea when a sealed envelope fluttered down to settle onto her mob cap. With a growl of annoyance, she snatched it off and looked around for any inerrant gaps in the spacetime continuum, but found none. She hadn't been expecting one, since there was no way Yukari would dally looking through one to see her reaction — or if she did, she would be sure to have it camouflaged.

Mumbling unkind things about the Great Troll of Gensokyo, Remilia ripped open the envelope to read the message within.

* * *

_Dear Remilia,_

_Before now, knight or knave, everyone you have encountered had beliefs coincident with reality, and therefore told the truth or lied according to that correct view of reality. Now you must consider that not only the being may be lying, but may not have the correct view of reality in the first place._

_Something strange has happened in the Forest of Magic — a peculiar madness has taken hold there, driving half the beings there insane. (I strongly suspect Lady Yukari's involvement this time, but I cannot say for sure.) Every being in this new Forest of Insanity is either sane or insane. A sane being believes only true propositions and does not believe false ones, while an insane being only believes false propositions and does not believe true ones. This is where the caveat from the first message comes in. Every being you have encountered so far is sane, at least insofar as this stipulated definition is concerned. Yes, even the powered-up Shizuha Aki was sane according to this definition: at best, the EX-ercise Machine only distorted her desires, not her beliefs._

_Sane or insane, someone cannot both believe some proposition and its negation. However, this does not apply to the logical consequences of these beliefs. Just as a (sane) knave can make logically incoherent statements, an insane person can believe logically incoherent statements._

_Sane knights and knaves have the familiar behavior, because their beliefs coincide with reality and can reliably tell the truth or lie about them. Insane knights make claims that are false, because they sincerely believe incorrect propositions and tell the truth about them (an insane knight can claim to be a knave, because she sincerely believes she is a knave, and as a knight tells the truth as she knows it). Insane knaves make claims that are true, because they sincerely believe negations of true propositions, then lie about the negations leading to a true statement (an insane knave can claim to be a knave, because she sincerely believes she is not a knave, but as a knave lies about this mistaken belief)._

_Natives of the forest are either sane or insane. Strangers to the forest such as yourself, however, are all sane. The cause of the insanity is unknown, but those who leave within a day or so upon entering have a reasonable chance of leaving the forest with their sanity intact. The longer you stay, however, the greater your chance of being driven insane. As your stay will be short, there should be no danger. On the off chance one of you are driven insane, however, I would ask that you write down your observations._

_Your objectives this time are Lily White, Letty Whiterock, Kogasa Tatara, Koishi Komeiji, Marisa Kirisame, and Alice Margatroid. You must not only determine the types of these people, but also their sanity — sane or insane._

_Be careful in dealing with the Seven Colored Magician. I've been hearing strange rumors about her as of late._

_Good Luck and Be Safe._

—_Ran Yakumo_

* * *

"That's _not_ how insanity works, Ran Yakumo! (I should know!)" Remilia declared as she finished reading off the letter aloud. At the time, she was standing in the library, over Patchouli (as best as she could, given her rather diminutive height). It was for her benefit that Remilia read the letter aloud.

Patchy, for her part, had calmly listened to Remilia's recitation of Ran's letter here in Viole. She then stated, matter of factly, "It's a stipulated definition, Lady Remilia. It overrides the usual meaning." Remilia supposed she had a bewildered look on her face, because Patchouli continued, "This definition of 'sane' and 'insane' comes straight out of Smullyan's book, and you should've reached that point by now, had you been keeping up with your reading."

Remilia gave an embarrassed grin. "Eh heh heh. Sorry, and you're right. I should have known that the Gap Hag would be using more of Smullyan's scenarios," she said, laughing off her failure to keep up with her studies. "Speaking of which, do you have any thoughts about that?"

"I have some observations. Ran leaves a lot unsaid in her letter. Implied by the statements she has given, but still, a great deal unsaid that will trap you if you are unwary," Patchouli stated evenly, suddenly taking on the air of a university lecturer.

"Oh?"

"For one thing, Ran hasn't out-and-out said that insanity reverses the truth value of statements made by knights and knaves," the unmoving magician began.

"Isn't that obvious?" asked Remilia.

"Why do you think it is?"

"If someone is insane, then their beliefs are reversed. An insane knight believes she is not knight, because otherwise her belief is correct and the insane cannot believe correct propositions. She must tell the truth as she believes it, and so claims not to be a knight."

Patchouli smiled. "You are correct that an insane knight does not believe she is a knight, because that belief is a correct one and —as an insane being— she cannot believe correct propositions, but I see nothing here that asserts that she does believe she is not a knight. What if she doesn't know either way, unlikely as that is?"

Remilia felt her face flush slightly with embarrassment. "Oh," she acknowledged Patchy's point in a very small voice.

"Furthermore, what if she's an insane knave? An insane knave does not believe she is a knave, because that belief would be correct and —being insane— cannot believe it," Patchouli continued, "but that doesn't mean she believes she is not a knave. See the problem now?"

"Yeah, I do," Remilia said with a frown. She had still much to learn about logic.

The magician continued her impromptu logic lesson. "Instead, we must go to one of the previous missives. Ran said that, in order to make a statement, the person must know the truth of what she's saying. She must know whether her statement is correct or incorrect — implying that she either knows the statement to be true, or knows the statement to be false. However, she has also stated that what I have put forward is a simplification of the actual situation.

"Everyone you have interviewed so far is sane. That is, their beliefs match with reality. So if we may refine that rule, it means that a person cannot make a statement unless they either believe the statement, or its negation. This collapses to Ran's simplified statement in the case of the sane people you've encountered thus far, because we may take belief in truth as knowing."

A light seemed to turn on in the loli vampire's head. "Ah. So we must actually wait until the person talks about something before we can conclude that they believe either way," she said in realization.

"Yes. Now, for a matter of externalities, like whether one is a knight or knave, what can we conclude about some statement, S, when said by either an insane knight or an insane knave?"

"Well..." Remilia hesitated as she thought it over. "An insane knight who says S seriously believes S is true, but being insane, S must in fact be false, so whatever an insane knight says is false. An insane knave who says S is lying about her belief in S, so she must seriously believe that not-S is true, but being insane, not-S is false, thus S is true, and therefore anything an insane knave says is the truth."

Patchouli smiled at her friend and student. "Spot on, Lady Remilia," complemented the librarian. "What you say is true for direct statements of fact, like whether one is insane or a knave, but what about statements of one's beliefs?"

"Um..." Remilia thought about this, but quickly got tangled on the beliefs of beliefs. "Oh dear, that might get complicated."

"Your caution will serve you well, I believe."

"Patchy, did you just tell a joke?"

Patchouli blinked, eyes slightly widening at the observation. Mentally reviewing her last statement, she supposed that, yes indeed, she had told a joke. "Not intentionally, and it's a very weak one at that," she admitted. "But back to the discussion. There may be some category of creatures where it is true that if he or she does not believe some proposition S, then they do indeed believe the proposition not-S. We may call such creatures _opinionated_ on S, and creatures who are opinionated on every S... simply 'opinionated.'"

Remilia remembered how Patchouli's handedness terminology caught on without having needed time to spread around. "I suspect that your 'opinionated' term is going to become very in-vogue by the time we arrive at the Forest," the lady of the house said dryly.

A ghost of a smirk crossed Patchouli's lips.

—/—

**Fruitcake Fairies in the Forest**

The Forest of Magic loomed ahead of the two travelers as dusk turned to night. Remilia closed her parasol that she used to shield herself from the sun's deadly rays as she traveled, as here in the forest, it was unneeded. This was not only due to the fact that it was night, but also because of the fact that the tree's healthy canopy would shield her in case they were caught by the sunrise.

"Well, let's get started," announced Remilia.

"It will not be immediately," lied Sakuya. "Where shall we start, mistress?"

"That thieving black-white ordinary human magician, with a name sounding like 'Melsia Beriberi,'" the young vampire declared.

"Her name is not Marisa Kirisame."

_61. Lily White is Our Guide?_

A white-clad fairy with a conical hat with red zigzag accents fluttered down on downy wings. She held up her hand in greeting, "Hello! Fancy meeting you again, Sakuya Izayoi!"

Sakuya jumped. She recognized this particular fairy! "Lily White? You're not here!" she gasped, then clamped a hand to her mouth as she blushed in embarrassment.

The Fairy Herald of Spring Lily White giggled, as Sakuya's statement had just exposed her as a knave, but said nothing. The Fairy of Spring sobered, opening a piece of paper she had been carrying. Remilia idly noted that an envelope was still safety-pinned to her clothes. "I got a message from the Great Troll's shikigami that I am to be your guide to the forest," she said, waving the letter about too quickly for anyone to read. She frowned at the letter, studying it intensely. "Don't know why she did that. It's not as if I'm completely up on everything that happens here."

"Are you sane, Lily White?" inquired the vampire.

Lily huffed indignantly, putting her tiny, white-bedecked arms on her tiny, white-bedecked hips. "Of course I am! I'm not a sane native of the forest, though."

Remilia knew then that Lily White could be trusted to guide them.

_Solution: Lily White claimed that she is sane. If she were a knave, then she actually believes she is not sane. If she were sane, then her belief is incorrect because she's actually sane and is not not sane, and sane people cannot believe incorrect propositions. But if she were insane, then her belief is correct because, being insane, she is indeed not sane, and insane people can only believe incorrect propositions. The assumption that Lily White is a knave leads us to a contradiction. Therefore, Lily White is not a knave. She is a knight._

_Because she is a knight and claims that she is not a sane native of the forest, she believes it. If she is insane, then she is correct in her belief that she is not a sane native of the forest, because she is an insane native of the forest. But this means her belief is true, and being insane she cannot believe true beliefs. Therefore, Lily is sane. Thus, her belief of not being a sane native of the forest is true. Since she isn't an insane native of the forest either, she is not a native of the forest; she is a stranger to the forest (and sane)._

—/—

"You're not from around here, are you?" Remilia observed.

Lily nodded curtly, but with a smile on her face. "Correct. In the off-season, I live just beyond the forest, quietly minding my own business and not attracting attention to myself," she said.

Remilia glared at her. "Considering all you do during the break of winter is draw attention to yourself, that sounds mighty strange coming from you."

"Don't confuse my job with my private life, Remilia Scarlet," Lily huffed indignantly. "Anyway, I didn't get hit with whatever magic drove many in the forest insane. All I felt was the wave of excess magic shucking off the forest," the fairy continued, "I went into it to find all my friends and find out whatever I could about what happened to them. I found in addition to the usual knight/knave shenanigans, this layer of insanity." Lily's face fell, a worried look settling about her. "It was so very difficult to deal with. I only managed to find out a few things about my friends, but sadly, there's still a bunch I don't know."

Lily White brightened as she held up her letter from Ran. "Ran said that you'll help me find out what happened to my friends! Thank you in advance, Lady Scarlet!"

Remilia felt slightly nervous. True, in solving these puzzles, Lily White would learn what happened to her friends, but it was apparent that Ran had told her an incomplete story. Helping her would be a side-product, not the main objective.

Lily continued. "I have quite a few pairs of twins that Ran says you'll help me with. The twins' names are as follows: Clearwater, Dewweed, Jasperstone, Humblebee, Gallflower, and Tumbledrum. Ran says that the sane knights of these fairies should be able to help you." She paused, studying the letter intently. "Um..."

"What is it, Lily?" asked the vampire youth.

"Lady Scarlet, what does 'opinionated' mean?" Lily asked, a confused look on her cute little face.

"What?"

"Ran's letter says that all of the fairies I just mentioned are 'opinionated,'" Lily clarified. "Do you have any idea what that means?"

Remilia let out an exasperated sigh and began explaining, not only the meaning of that particular word, but of their situation in general...

—/ _Girls are Explaining..._ /—

"Un. I see! So either Lady Yakumo or her shikigami has set various tasks in front of you to solve this mess!" Lily White said in understanding. "Then Ran did not tell me to guide you just so that you could help me find out what happened to my friends, as I had assumed."

"Very astute of you, Herald of Spring."

"Heh heh!" Lily giggled, looking embarrassed and proud at the same time. She snapped out of this and a determined look settled upon her features, steeling herself to the task ahead. "Very well! If helping you helps me, then I'll be glad to do so! I shall tell you what I have learned about my fellow fairies!" she said boldly, staring out with the first fact she had learned: "In the forest, both of a pair of a fairy twins cannot both be knights."

"Outside the Forest, fairies of such a pair cannot both be knaves," countered Remilia. "Is that still true?"

Lily nodded vigorously. "Yes. So, a pair of fairy twins is composed of a knight and a knave — at least, when they're natives of this forest."

"This actually might make things easier," Remilia mused. After all, there would be only two possibilities to consider instead of three.

Lily paused to read the letter further. "It says here that for some of the fairy twins, the knight will be sane. We are to bring along the sane knights, and will help us in the next phase of your journey," Lily explained.

Remilia frowned at that. "Seems the Gap Hag chained the problems again," mumbled the vampire sourly as they approached the first pair of fairy twins.

_62. Onyx and Beryl Clearwater_

"These two are Onyx and Beryl Clearwater, and they're twins," Lily said, gesturing at each of the identical twins in their turn. Unlike in the Scarlet Devil Mansion, each fairy was wearing a dress that suited their own taste, and so could be used as a means to distinguish them. "I'm afraid I don't know anything else relevant to your quest about them," Lily continued, somewhat subdued from her original introduction.

"It doesn't sound like you really tried."

"I'm just a fairy! Cut me some slack!" whined the fairy herald.

Shaking her head, Remilia confronted the twinned fairies. "Onyx, Beryl, would you mind telling me about yourselves?"

One twin, Onyx, replied first. "We are both insane."

Beryl, the second twin, countered the first, "That's not true!"

"I am a knight," Onyx added.

Remilia knew then both their types and sanities.

_Solution: Onyx's statement is either true or false. If it is true, then Onyx is insane, and the only insane inhabitants of the Forest who speak truly are knaves. Thus, Onyx is the knave and Beryl is the knight (both insane). If Onyx's statement is false, then at least one of them is sane. If Onyx is sane, then she has lied, and so she is the knave, and Beryl is the knight. If Beryl is sane, then has spoken truly in contradicting Onyx, and so must be the knight, making Onyx the knave (and sane). In all cases, Onyx is the knave and Beryl is the knight._

_Onyx is a knave, so by her second statement, Onyx doesn't believe she is a knight, and being opinionated, she therefore believes she is not a knight. This is true, for she is a knave. Therefore, she believes something that is actually true, and she is thus sane. Beryl is a knight, and therefore she really believes that the twins are not both insane, and of course, we have just shown that Onyx is indeed sane, so her belief that they are not both insane is true. Because Beryl believes a true belief, she is also sane._

_63. Honey and Faith Humblebee_

Remilia directed Beryl Clearwater to follow them, as she bore information for the next part of Remilia and Sakuya's quest.

The tour of four persons proceeded through the forest, Lily White leading the way, until they happened upon another pair of fairy twins. Lily smiled at the sight of them. "These are Honey and Faith Humblebee. I know nothing about them relevant to your quest, either."

Remilia let out a sigh of exasperation. "Fine. Honey and Faith, tell me something about yourselves."

"I am a knight," said Honey.

"I am a knight, too," said Faith.

Honey nodded towards Faith. "My sister is sane," she claimed.

It didn't take long for Remilia to figure out their sanities and types.

_Solution: Suppose Honey really is a knight. Then she believes she is a knight, and because she actually is a knight, she is sane. If Honey is a knave instead, then she believes she isn't a knight and is lying about it. But she actually is not a knight, and so is also sane. Therefore, Honey is sane. The same argument applies to Faith, and she is also sane._

_Honey believes either that Faith is sane or Faith is insane. But of course, Faith is sane, as is Honey, and therefore Honey cannot believe Faith is insane, and must believe she is sane. Since this is exactly what she claims, Honey is a knight, and therefore Faith is a knave._

_64. Lyra and Bonnie Dewweed_

Honey Humblebee joined Beryl Clearwater on their troupe as hangers on.

A bit further into the forest, Lily stopped and pointed towards another pair. "Here are Lyra and Bonnie Dewweed. Like the others, I haven't figured out their sanities."

"If you will..." Remilia prompted.

"I am a knave," said Lyra.

"I am a knight," Bonnie said.

"Bonnie and I are the same as far as our sanity goes," Lyra added.

Remilia knew then she was wasting her time with these two.

_Solution: Lyra claims to be a knave, which makes her insane. For a knight claiming to be a knave must believe she is a knave, which is not true and thus makes her insane, and a knave that makes such a claim must believe she is not a knave, which is again not true and thus makes her insane. Bonnie claims to be a knight, which makes her sane (as we found in the previous problem, a person claiming to be a knight is sane). Therefore, the two are not alike as far as their sanity goes._

_This contradicts Lyra's seconds statement: contrary to her claim, Lyra and Bonnie do not match in sanity. Since Lyra is insane and uttered an untruth, she must be the knight. This makes Bonnie the knave, who is also sane._

_65. Ruby and Amber Jasperstone_

Neither Lyra nor Bonnie Dewweed joined the troupe, being unsuitable for Remilia's purposes. A few minutes further and they came upon yet another pair of twins.

As Lily spied this third pair, she clapped her hands together with excitement. "Ah! This pair I know something about!" she said happily, "These are the fairy twins Ruby and Amber Jasperstone! I managed to figure out that one of them is sane and the other is insane!" Lily sounded quite proud of herself.

Remilia scratched the back of her head, completely disinterested in Lily's self-congratulation. "So," she said, addressing the two fairies, "Do you want to tell me about yourselves?"

"My sister is a knave," Ruby said, pointing at Amber.

"My sister is insane!" Amber roared.

Remilia was about to leave them behind on sheer principle, but she found a better justification.

_Solution: Suppose Amber really is a knave. Then Ruby is the knight, and furthermore, must be sane, as she's told the truth. This means that Amber, being a knave, has lied and actually believes her sister to be sane, which is true. But this implies that the both of them are sane, which puts lie to Lily's statement, and we already know she is a sane knight. Therefore, Amber cannot be the knave; she is the knight._

_This makes Ruby's statement false and makes her the knave. Therefore, she doesn't really believe that Amber is a knave, and being opinionated she must believe that Amber is not a knave, which is true: Amber is a knight. Therefore, Ruby is sane. But Amber claimed that Ruby is insane, and she is a knight, therefore she really believes Ruby is insane, which is wrong, and therefore Amber herself is insane._

_66. Viola and Rosie Tumbledrum_

Again, no one joined the train of fairies.

"We're picking up fairies very quickly," observed Sakuya in her knavish manner.

"Indeed, we are not. With any luck, that will make any puzzle connected with getting all of them a snap," Remilia speculated. "Speaking of which, it looks like we're coming up on the next pair."

"This is Viola and Rosie Tumbledrum!" Lily said enthusiastically, "Like the Jasperstone twins, one of the Tumbledrum twins is sane and the other is insane." The fairy turned to the twins, and instructed them, "Say something about yourselves, girls!"

"I am a knave," Viola boldly claimed.

"If that were true, then I am a knight," Rosie added.

Remilia considered this one easy.

_Solution: Since the two fairies differ in both type and sanity, they either both tell truths or both falsehoods. For if the knight is sane, the knave is insane, and thus both tell truths, and if the knight is insane, the knave is sane, and so both tell untruths._

_If Viola really is the knave, then Rosie is most certainly the knight, which means that Rosie's statement is true. Since Rosie tells truths, by the above derivation, they both must do so, so the knight must be sane and the knave insane. Viola is definitely insane — no sane being can claim to be a knave. Therefore, she is the insane knave and Rosie is the sane knight._

_67. Petal and Posey Gallflower_

Rosie Tumbledrum joined Honey Humblebee and Beryl Clearwater, being suitable to Remilia's future task.

"These are Petal and Posey Gallflower," said Lily, introducing the twins.

"You wouldn't happen to know anything about them, would you?" the vampire asked the spring herald.

Lily, predictably, shrugged. "Sorry. I don't know anything further about them," she admitted. "Please tell us something about yourselves."

Petal started. "I am either a knave, or a native."

"Petal is an outsider. She is also a knave," Posey contradicted.

Lily clapped her hands together in a flash of realization. "Hey! I figured it out!" she shouted.

Remilia gaped. "What? How did you—?"

"I haven't just been floating around being a convenient plot device, you know! I've been learning!" Lily said with a grin.

_Solution: If Petal's statement is false, then she is neither a knave nor a native. She must then be an outsider knight, and if she is an outsider, she is sane, so she is a sane knight. But sane knights tell the truth, so her statement cannot be a lie. Therefore, she is either a native or a knave (or both). Because the statement is the truth, Petal is either a sane knight or an insane knave. If she is a sane knight then she is a native because she is not a knave; if she is an insane knave, then she is a native because she is insane._

_Posey's first statement is false, because Petal is indeed a native — Posey has lied or believes one. The twins therefore must be of equal sanity (see previous puzzle). Since they do not both speak the truth or speak wrongly, but are definitely of different type, they cannot have different sanities._

_Posey's second statement is also false, because Posey has established herself as someone who tells untruth. Therefore, Petal is a knight, and therefore a sane knight because she tells the truth. This establishes Posey as a sane knave, because the two have equal sanity, and opposite type._

—/—

**More Fairy Family Relationships ~ Fairy Cousins**

Petal Gallflower was the final addition to the entourage now consisting of herself, Rosie Tumbledrum, Honey Humblebee, and Beryl Clearwater. All of them were sane knights.

Remilia confronted the four fairies. "Alright, you four. You all are sane knights. What is this mysterious help that Ran Yakumo expects you to provide me?"

Beryl Clearwater spoke first. "There are are four pairs of fairy cousins we must find: Mary and Gertrude Pennypease, Carol and Tammy Dandelion, Penelope and Catherine Caneriver, and Halley and Jade Dallywinds," she said. "We each have one fact to reveal about these pairs of fairy cousins. My fact is that each sane fairy from these pairs has a key that will be important later and will be given to you upon request, although we are not allowed to tell you much more about what the keys are or what they do."

"Fairies have cousins?" Remilia asked, her surprise catching her off guard.

"Yep, they do," nodded Lily White, the fairy knight.

Remilia narrowed her eyes. "And I assume that being cousins implies something about they being knights and knaves, or sanity or insanity?" prompted the vampire.

Lily nodded again. "Yep. In this forest, fairy cousins are the same type: they are either both knights, or both knaves."

Remilia realized immediately what that meant. "More pointless puzzles. This is just blatant busywork."

_68. Mary and Gertrude Pennypease_

Lily flew up to two pink-haired fairies, a small one with a chubby face, and the other tall with a thin, regal face. "These two are Mary and Gertrude Pennypease!"

"There is nothing I need tell you about those two," Beryl added.

"Typical. Let's get this over with," Remilia grumbled. "You two. Tell me about yourselves."

Mary, the chubby-faced one, said, "My cousin is a knave."

Gertrude, the tall, regal faced one, said, "My cousin is a knight."

Mary added, "One of us is sane and one is not."

Remilia figured out who held the key and collected it.

_Solution: Exactly one of the two cousins must have told the truth, because they are either both knights or both knaves. They cannot have equal sanity because then they would match in their ability to tell the truth, which they don't. Thus, Mary is telling the truth, and Gertrude an untruth. Therefore, they are both knaves. Since Mary is a knave who tells the truth, she is insane. Thus, her sister is a sane knave, and has one of the keys._

_69. Carol and Tammy Dandelion_

Remilia regarded the gold (or perhaps brass) key with a diamond shaped bow — 'diamond' as in the card suit, and the bow even had a red gem that might have been a ruby in its center. It looked like an ordinary ward lock key, but knowing its probable source, it probably unlocked some cosmic horror, or maybe a tea chest full of really tasty tea.

"No pair of fairies in sight," quipped Sakuya, then pointed towards the obvious pair of fairies right ahead of them.

Lily buzzed up to the pair. "These are Carol and Tammy Dandelion!" she said.

Rosie Tumbledrum pulled up besides Lily White. "I can tell you something of these two: the both of them are opinionated," she said, breaking into Lily White's introduction.

The herald of spring was pretty put out by this interruption. "Hey! That's rude!" she squeaked angrily, windmilling her arms.

"Yes, it was," Rosie said with a smile.

_Of course they'd agree,_ thought Remilia, _They're both sane knights._

The vampire turned to the two. "Enough of this tomfoolery. Tell me something about yourselves, you two," she curtly ordered the fairies.

"Whatever my cousin says is true," said Carol.

"My cousin is insane," added Tammy.

Remilia puzzled on that for a minute before asking, "Tammy, do you believe you are sane?"

"Yes," came her answer.

Remilia collected all the keys the two had.

_Solution: Suppose Tammy really does tell the truth. Then Carol is either a sane knight or an insane knave. But if Tammy tells the truth, then Carol is insane. Therefore, Carol is an insane knave. Suppose Tammy tells falsehoods. Then Carol is either an insane knight or a sane knave because her statement is incorrect, and because Tammy tells falsehoods, her claim means that Carol is actually sane, and because she tells lies Carol must be a sane knave. Therefore, depending on whether Tammy really is telling the truth or not, Carol is either a sane or insane knave, and Tammy is also a knave of the same sanity level._

_By her answer to Remilia's question, Tammy believes she does not believe she is sane. This implies that she is insane. For if Tammy is sane, then she really doesn't believe she is sane. Because she is opinionated, Tammy then believes that she is not sane. But this is not correct: she really is sane, and she believes a falsehood. Thus, if she is sane, she is insane. Therefore, Tammy must be insane. This makes her an insane knave, and therefore she really does believe she is sane (but of course, this belief is wrong, being insane). Because Tammy is insane, Carol is too._

_Because each of them are insane, neither has a key._

_70. Penelope and Catherine Caneriver_

"Lady Remilia, look!" Lily shouted as she dragged two fairies to the group. "These are my best friends, Penelope and Catherine Caneriver," bubbled the Fairy Herald of Spring, introducing the two in turn. "Tell Lady Remilia something about yourselves, girls!"

Penelope, a fairy with pink hair, said, "We are both insane."

Catherine, with blue hair, nodded in agreement. "That is true."

"I understand," Remilia replied. "Are you a knave, Penelope?"

"Only if my sister is as well!"

Their statements revealed to Remilia who of the two had a key.

_Solution: The two cousins agree that the both of them are insane, and since they share the same type, they must also share the same sanity: if one is insane, both are insane. If they are both sane, then they cannot believe they are both insane, because that belief is false. If they are both insane, then they cannot believe they are both insane, because that belief is true. Therefore, neither believes they are both insane. Penelope and Catherine therefore believe the opposite of their claim and are therefore both knaves._

_Penelope claimed that she was a knave only if her sister is. Of course, they are cousins, so if Penelope is a knave, then Catherine is also a knave, so her statement is true. She is thus a knave who tells the truth — an insane knave. And of course, if she is insane, Catherine is too. And since they're both insane, neither was in possession of a key._

_71. Halley and Jade Dallywinds_

"May I introduce to you Halley and Jade Dallywinds!" said Lily, throwing out her arms with joy. Remilia shook her head.

"You will now tell us something about yourselves," Beryl said to the cousins. It was not a command, but a statement of fact.

Halley, a fairy with light brown hair, said, "At least one of us is insane."

Jade, with light green hair, retorted, "That's not true!"

"We are both knights," Hally added.

Remilia figured out who had a key and collected them.

_Solution: Halley and Jade disagree with each other, and because they are the same type, they must be differing in sanity: one is sane, the other is insane. This means Halley is telling the truth, so she is __telling the truth when she says they are both knights. Therefore, Halley is a sane knight and has a key, and Jade is an insane knight._

_72. Pipi_

Remilia took the ward lock key with the heart-shaped bow from Halley and continued on their way. She noted that the heart key had a red gem in its bow like the diamond key.

The next fairy they encountered was a small, lone fairy, easily the smallest Remilia had ever seen. (Not that she'd seen many. Remilia would have to ask Reimu about fairy sizes one day.) She was unaccompanied.

Suddenly, the fairy shouted, "I believe that I have both a cousin and a twin!"

Remilia was taken aback by Pipi's sudden outburst. "What?" she asked, but the fairy had already skedaddled away.

Lily smiled gently. "Pipi is rather shy," she said.

Honey laughed, "I also know Pipi. She is a native of the forest and opinionated on that particular belief."

Remilia had figured out one thing about Pipi.

_Solution: Pipi is a knave._

_Pipi could not have both a cousin and a twin. For she would be of opposite type to her twin, and her cousin would be the same type as her, and at the same time be the same type as her twin (because her cousin is cousin to her twin as well as to herself). Thus, the fairy's twin would be of both the same type and of opposite type to herself, which is impossible. Therefore, Pipi (or any fairy of the forest) cannot have both a cousin and a twin._

_Suppose Pipi really does believe that she has both a cousin and a twin. Then she has told the truth, because this is exactly her statement. On the other hand, she must be insane, because this belief she holds is surely false. The only individuals in the forest that are insane and speak true statements are insane knaves._

_On the other hand, suppose Pipi doesn't believe that statement. Since she is opinionated on whether she has a cousin and twin, she must believe that it is not the case. Since it cannot possibly be the case, her belief is true and she is sane. She cannot believe that she does believe the statement, because she in fact does not believe it and sane people do not believe falsehoods. But she has made a statement about it, so she in fact believes she does not believe the statement. Again, she has lied about her belief, so Pipi must be a knave! We cannot tell if Pipi is sane or insane, however._

_73. Pinkie Applebloom_

A fairy wearing a pink dress and having a head full of pouffy maroon hair wove her way erratically through the trees and the flowers like a drunken bumblebee. She was so atypical that Remilia would have bet money that she was involved in their quest.

"Do you know anything about this fairy?" she asked Lily, pointing at the unusual fairy.

"She goes by the name Pinkie Applebloom, and she's a strange one."

"That's a matter of opinion," Remilia pointed out.

Lily exploded, flapping her arms in a tizzy of excitement, as she hastily explained why Pinkie was a nut (though not necessarily insane). "You don't understand! She's totally random! Even before the current incident, she was an enigma filling, wrapped up in perplexity bread, sauteed in a bewilderment sauce and garnished with imponderable parsley, served with a side of conundrum."

"Surely, you're exaggerating," said flatly. The logic sense in her nagged, however. Sane knights like Lily were not prone to exaggeration, as it was a form of lying. Nevertheless, she ignored it and addressed Pinkie Applebloom directly. "Pinkie, please tell me about yourself."

"I believe that I am a knave!" Pinkie announced, quite loudly. She also struck a pose that would not look unusual on Iku Nagae.

_Disco feevahhh!_ a rogue thought floated through Remilia's mind, as she searched for a way to answer her. "...I see. Pinkie, are you—"

Pinkie cut her off. "It is not the case that I am opinionated if I am a knight. Also, I have a key." With that, the fairy flew off, leaving a bewildered troupe of Gensokyoites behind.

Lily grabbed her head in frustration. "Grr! What did I tell you! Completely random! I didn't figure out anything about her!" she raved. The fluttering of her wings increased and she began to ascend. "I'll chase her down!"

Remilia casually plucked her out of the air before the fairy herald could pursue. "On the contrary, Lily, I've figured out _everything_ about her."

_Solution: This one is tricky! Believe it or not, Pinkie is a sane knave, and she is unopinionated!_

_Suppose Pinkie really believes she is a knave. If she's a knight, then she is insane, because she's not really a knave — she is a knight. But insane knights do not tell truths, and this fairy has claimed exactly that she believes she is a knave. Therefore, she cannot be a knight, but a knave. Because she does believe she is a knave, which is correct, she must be sane. But then she has told the truth about her believing she is a knave, which a sane knave cannot do. Therefore, Pinkie cannot really believe she is a knave._

_Pinkie's statement is therefore not true; she does not believe she is a knave. She can only be an insane knight or a sane knave. Therefore, her statement that it is not the case that she is opinionated if she is a knight is also not true: if she is a knight, she is indeed opinionated. Therefore, if she is a knight and she doesn't believe she is a knave, then she believes she is not a knave. This is true, so she is sane. But this would make her a sane knight, and she can only claim to believe she is a knave if she is an insane knight or a sane knave and she is neither, so she cannot be a knight at all. Pinkie must be a knave. Since we have already shown that Pinkie must be an insane knight or a sane knave, and she is not a knight, she must be a sane knave._

_But if Pinkie is a knave and doesn't believe that she is a knave, then would that imply that she is insane? No, only if she is opinionated (in particular, she is opinionated over whether she is a knave), for which not believing she is a knave is equivalent to believing she is not a knave, which would make her insane. However, she is not necessarily opinionated if she is a knave (only if she is a knight), and if she were opinionated, she could not be a knave either. Therefore, she is unopinionated (in particular, not opinionated on whether she is a knave) and therefore neither believes she is a knave nor believes she is a knight. (She must have missed the memo!)_

_And finally, since she's a sane knave, she's lying about having a key, so she has no key._

—/—

**Party of Four**

"And so Pinkie is a sane knave who is unopinionated! That's neat!" Lily exclaimed giddily and clapped as her head finally wrapped completely around Remilia's explanation of how she figured that fact out.

"Thank you," Remilia replied, preening a bit. Sure, the praise of her intellect was from a fairy, a race not known for its smarts, but praise was praise. Remilia realized this quickly and did not preen long, knowing how pathetic it might seem, so she moved onto the next topic, "Those two encounters seem rather out of place. Neither of them had a cousin we could talk to."

"Hmmm..." mumbled the white-bedecked fairy.

A moment passed before Sakuya coughed politely, holding up an envelope. "This letter did not drop from the fairy Pinkie Applebloom's robes," she lied.

"Ah. That's the explanation. Alright, let's see what Ran has to say."

* * *

_Dear Remilia,_

_One of your fairy twin companions is the bearer of a secret that will allow you to proceed with your quest. Sadly, only Lily White and but one of your fairy twin companions are allowed to accompany you and Sakuya further. Here you must make a choice of which will accompany you — after you make your choice, the rest of the fairies will disappear, one way or another. It is advisable that you choose the fairy who is the bearer of the secret I have mentioned._

_But which to choose? That is the challenge! You are allowed to ask one and only one question of a single fairy twin you have accumulated. You are to make your decision on the basis of her answer. All these fairies will know who the secret-bearer is. However, they have been instructed not to directly reveal any but their own status as the secret-bearer. If you ask the identity the secret-bearer by name or any similar question, they will refuse to answer and you will have wasted your only question._

_Petal Gallflower bears two clues that will help you. A little thought will show you that what I have given you, these two clues, and your single question will be sufficient for you to find the secret-bearer._

_Good Luck._

—_Ran Yakumo_

* * *

"Confound these puzzles!" Remilia growled.

_74. Who has the Secret?_

Petal Gallflower flew up to Remilia, knowing what was in the letter, and gave her a few clues. "The first clue mentioned in Ran's letter is that the secret-bearer is an outsider to the forest. The second is that if a fairy is an outsider, so is her twin."

Ran was right. A little thought _did_ give her enough information to formulate a question that would tell her all she needed to know. She directed her question to Beryl Clearwater, and from her answer, Remilia was able to tell who should be her forth companion.

_Solution: Many questions will work, but the one Remilia asked of Beryl Clearwater was the simplest: "Are you the secret bearer?"_

_Petal Gallflower established that fairy twins are either both native to the forest or both outsiders, and that the secret-bearer is an outsider. Furthermore, Petal Gallflower had established herself as a native with her statement in a preceding problem, and Rosie Tumbledrum's sister, Viola, is insane, and therefore a native, so Rosie is also a native. This leaves Beryl Clearwater and Honey Humblebee as the the possible secret bearers, as only they can be outsiders._

_Ran's letter says that Remilia's question is within the rules, so Beryl will answer. If Beryl claims to be the secret bearer, then as a sane knight, she is indeed the secret bearer. If she claims no such thing, then she is telling the truth and she is not the secret bearer, which implies that someone else is. The only other possibility is that Honey is the secret bearer._

—/—

Beryl had answered Remilia in the negative; that she was not the secret bearer. Remilia immediately pointed to Honey Humblebee and announced, "I choose you, as you are the secret bearer."

Honey nodded as she bowed to Remilia. "Excellent choice, Lady Scarlet. You have chosen wisely," she said as the other fairies dispersed. Remilia let them go, as Honey was the only one she needed, and she was staying.

"I realized pretty quickly that I should be able to logically exclude two of your compatriots, narrowing the field to you and Beryl," Remilia explained. "Ran's challenges are difficult, but any more possibilities would make the challenge unfair."

Honey nodded with a smile. "Very nice insight. But now we need to find Letty Whiterock, as per your instructions," she said with a smile.

Letty Whiterock was not hard to find. She was sitting on a rather large boulder, in the center of a large cold spot in the forest, humming to herself.

"I would have thought that you and Letty would only appear in different seasons," Remilia remarked to the herald.

Lily frowned. "Who told you that? I'm only the _herald_ of spring, not its essence. I exist in other seasons, too, as I am doing now," the fairy herald said.

Remilia thought back to the Aki sisters they had met just a couple of days ago. "I suppose that's true, given the Aki sisters," she remarked aloud, "But how do you get back to the west without anyone seeing you?"

"Hee hee! Fairies' Secret!"

Remilia felt a touch of annoyance being patronized by a fairy. Still, she tried not to let any annoyance seep into her voice. "That still doesn't answer my question: how can Letty appear out of her season?"

The fairy herald shrugged. "I donno. Maybe, like the heralds, the sprits of the seasons also appear in seasons different from their 'assignments.'" Lily quoted 'assignments,' as the seasonal duties weren't actually assigned as such.

"Aren't you supposed to know this, Herald?"

"Why should I? I just _announce_ spring. I don' bring it."

_76. What Winter Left Way Behind_

"Letty Whiterock is the Spirit of Winter. As a seasonal spirit, that means she knows which season it is at all times, or rather she _should_," Letty began, holding up a lecturing finger. "Every seasonal spirit I've encountered, sane or insane, believes exactly one season is the correct one. The same will be true of Letty."

"Are you sure?"

Lily shrugged again. "I don't know how to explain it, but it is one thing I am dead certain of. No doubt in my mind at all," she assured.

"More implanted memories."

"I guess," Lily agreed. "Anyway, it's important that a seasonal spirit knows which season it is. If they don't have their seasons right, we'll have a struggle between seasons, as two seasonal spirits try to assert their season, or a season without a season, as it were."

"I suppose that's bad."

"Very! You get crop failures and poor harvests and humans get really really rowdy after that."

Remilia drew back, looking strangely at the spring fairy. "And how would you know about crop failures?" she asked, genuinely curious.

"Hey, I've been around long enough to know that humans need food and farming is how they get a good chunk of it. I also know that spring is the time when they start planting." Lily's eyes were narrowed at the vampire. "'M not _completely_ dumb, ya know," she insisted.

Remilia huffed, "I never said that." The vampire turned to Letty, who was standing passively where they found her, not making a sound. "You've been pretty silent, Letty, so answer me this question: Yes or no, are you either a knight who will answer 'no' to this question, or a knave who will answer 'yes'?" Remilia asked at last.

Letty gave her answer, and it was one of 'yes' or 'no.' Her answer made Remilia grimace with consternation. "Ladies, if this Incident continues, we may be in for some problems," Remilia said aloud, an image of an asesonal winter firmly in her mind.

_Solution: By the very fact that she answered the question, either 'yes' or 'no,' Letty has proven that she is insane!_

_If Letty is a knight and answers 'no,' then she is a knight who answered 'no' to Remilia's question. This makes the question's proper answer 'yes,' which means that Letty's answer is wrong. If Letty instead answers 'yes,' then she is not a knight who answered 'no' to the question, neither is she a knave who answered 'yes' to the question (since she isn't a knave at all), and therefore Letty's answer is still wrong. There is no way that Letty can correctly answer the question, 'yes' or 'no,' if she is a knight, so since she does answer, she must have answered wrongly and must be insane._

_If Letty is a knave and answers 'yes,' then she is a knave who answered 'yes' to the question. This makes the question's correct answer 'yes,' which means that her answer is correct. If Letty instead answers 'no,' then she is neither a knight who answered 'no' (since she is not a knight), nor a knave who answered 'yes,' and therefore the correct answer is 'no,' which is her answer. There is no way that Letty can answer incorrectly, 'yes' or 'no,' if she is a knave, so since she does answer, she must have answered correctly and again must be insane._

_Thus, knight or knave, if Letty answers the question at all with a 'yes' or a 'no,' she is insane! (The only way a sane inhabitant can escape this if she refuses to answer!)_

—/—

**The Ghostly Umbrella and the Two-Eyed Satori**

"So, we've discovered that Letty is a nutter, and we have in our possession a pair of keys of untold function, and a pair of sane and knightish fairies," summarized Remilia as the group made their way deeper into the forest. "I wonder what these keys are for."

Lily shrugged her shoulders, and Honey simply remained silent on this point.

"All we can do is demand answers now," stated Sakuya, falsely.

_"BOO!" _shouted the humanoid half of the youkai known as Kogasa Tatara, the Cheery Forgotten Umbrella.

Neither Remilia nor Sakuya jumped in surprise, but Kogasa found that Sakuya's reflexes were well honed, as a knife embedded in her forehead.

"Ow," Kogasa said, pained as she extracted the knife. Whether this was because she was a youkai and was just that tough, or because the humanoid body wasn't her actual self, was anyone's guess.

"Hello!" chirped Koishi Komeiji, the Closed Eyes of Love, unconsciously as she simply appeared out of nowhere.

"Ah, Kogasa Tatara and Koishi Komeiji," said Honey Humblebee aloud. "I've been expecting you, as we have business to square away with you. Or, more accurately, Lady Remilia Scarlet has business to square away with you."

"Usual rules?" asked Remilia.

"Usual rules," confirmed Honey.

_76. First Round of Statements_

"I suppose I'll start with this well-worn instruction: please tell me about yourselves or your companion," Remilia began.

Kogasa smiled, saying, "I believe Koishi is sane."

"Kogasa is a knight," added Koishi.

Honey added in her own hint, "These two are opinionated on their types and sanities."

Remilia thought about it for a minute, then announced, "This isn't nearly enough information. I can only figure out Koishi's type!"

_Solution: All we can tell at the moment is that Koishi is a knight._

_Suppose Kogasa is telling the truth, and she really believes that Koishi is sane. Then Kogasa is either a sane knight or an insane knave. If she is the former, then since Kogasa is sane, it is true that Koishi is sane. This also makes Koishi's statement true, since Kogasa really is a knight, so Koishi must be a sane knight. If Kogasa is the latter, then her belief that Koishi is sane implies that it is false, so Koishi is actually insane. This also makes Koishi's statement false, so she must be an insane knight. Thus, if Kogasa really believes Koishi is sane, then Koishi is a knight, and has sanity equal to Kogasa._

_Suppose instead that Kogasa is not telling the truth, and she doesn't really believe that Koishi is sane. Then Kogasa is either an insane knight or a sane knave. In addition, Kogasa is opinionated on Koishi's sanity, so she either believes that Koishi is sane, or that Koishi is insane (not sane). She does not believe that Koishi is sane, she must believe that Koishi is insane. If Kogasa is an insane knight, then her belief that Koishi is insane is not true, so Koishi is sane, and because Kogasa is indeed a knight, Koishi tells the truth, so she is a sane knight. If Kogasa is a sane knave, then Koishi is indeed insane. Also, Koishi has told an untruth about Kogasa's type, so Koishi must be an insane knight. Thus, if Kogasa doesn't really believe Koishi is sane, then Koishi is a knight, and has opposite sanity to Kogasa._

_In all cases, Koishi is a knight, and Koishi and Kogasa match in sanity if and only if Kogasa really believes Koishi is sane._

_77. One Question_

Having only figured out one thing about one of the two, Remilia decided on her follow-up. "Is Kogasa sane?" the loli vampire asked the satori.

"I believe so," came her answer.

"Well, I don't!" countered Kogasa.

Remilia smiled. She now knew exactly what each of them were.

_Solution: Koishi claims, not that Kogasa is sane, but that she believes Kogasa is sane. Because she is a knight, she believes she believes Kogasa is sane. If Koishi herself is sane, then it is true that she believes that Kogasa is sane, and hence, it must be true that Kogasa is sane. If Koishi is insane, then it is not true that she believes Kogasa is sane. But she is opinionated on Kogasa's sanity, so she believes Kogasa is not sane, but since Koishi's insane, this belief is false, so again Kogasa is sane._

_Kogasa denies she believes she is sane. But she is sane, so she cannot believe she is not sane. Since she is opinionated on her sanity, she therefore believes she is not not-sane, or sane. She therefore does not speak the truth, but is nevertheless sane by Koishi's answer. Therefore, she is a sane knave and she lies._

_Because she lies, she doesn't really believe Koishi is sane (see previous solution), and we have already deduced that if that is the case, then Koishi is of opposite sanity to Kogasa, and hence insane._

—/—

**The Kirisame Magic Shoppe**

Remilia marked off a couple of ticks on her To Do list, now that she has found the complete types of both Kogasa and Koishi. "Okay, that leaves the Broom-Riding Thieving Intruder and the Seven-Colored Puppeteer," she stated.

"Marisa's place is not around that bend," Sakuya pointed out, breaking silence once more. Remilia noted that her maid had become a woman of few words since the incident began, owing to her now knavish nature. Sure, the maid had some fun early on, but seemed to have tired of that quickly and fell mute. Remilia felt sort of sorry for her, but there was nothing to be done but to resolve the incident, and that meant doing things Yukari's way for now.

Marisa's place was sited in the clearing that they just emerged from, and 'place' was about as truthful a way to characterize the domicile without getting insulting. As always, it was characterized by its poor structure. Looking at it, it was easy to believe that Marisa would not emerge for days from it, because it was likely that the mushrooms that she was rumored to eat regularly would grow on the inside of the structure, using the very members that the structure was built from as nourishment.

Remilia was in no mood to laugh at the hilarious structure, even though the urge was definitely there. She instead strided up to it, Sakuya hurrying after her to hold the parasol over her head. With one pale hand, she knocked smartly on the door, which seemed more solid and sound than it appeared.

The door opened, revealing the blonde-framed face of Marisa Kirisame. She was also framed by a few lengths of bandages around various limbs and her head, covering one eye, and her left arm hung limply in a sling. In short, in comparable condition to the structure she lived in. "Yo, Remilia! What can I do you for?" she greeted the vampire and maid with a wide, but pained grin on her face.

_78. The Ordinary Magician and the Fairy_

Remilia cut right to the chase. "I want answers to two questions: First, are you a knight? And second, are you sane?"

Marisa stared at Remilia, flatly and without answering. Remilia could tell that she had been asked these two questions before. Was she quizzing herself whether or not to answer the questions? Then Marisa grinned, seeing Honey Humblebee. "_She_ knows those answers," she replied, pointing at the companion fairy.

"C'mon! Just tell me, dangit!" Remilia snapped.

"Do I look like the sort of person who would wantonly give out easy hints?" she asked.

Remilia looked into the black-white's eyes, crinkled almost shut by the wide grin just below it. "No," she answered simply. She turned to Honey Humblebee, "Well? Do you know whether or not Marisa's a knight or knave, sane or insane?"

"I'll only say that Marisa is not an insane knave. You also have quite enough information to know the type and sanity of Marisa," Honey replied with a strange smile on her face.

Remilia opened her mouth to protest that it wouldn't be nearly enough, but clamped her mouth shut. "I am faced with two incredible facts, and I mean 'incredible' in both senses of the word," Remilia remarked after thinking about the matter carefully.

_Solution: Marisa is a sane knight, and Honey Humblebee is omniscient!_

_Someone is omniscient if they know all propositions to be true or false. Someone knows a proposition is true or false if that person believes that proposition if and only if that proposition is true._

_Take any proposition, S. Since Honey is sane, if she believes S, then S is true. Conversely, if S is false, she does not believe S. Furthermore, she is opinionated, so for every S, she believes exactly one of S or not-S. If S is true, then not-S is false. Therefore, Honey, being sane, does not believe not-S. But if she does not believe not-S, then she does believe not not-S (because she is opinionated), and therefore believes S. Thus, if S is true, Honey believes S._

_Honey therefore believes S if and only if S is true. Since this applies to any S, for all propositions, Honey believes that proposition if and only if it is true, and any being that fulfills this criterion is omniscient._

_Marisa is not an insane knave, because Honey said so, and she is omniscient and a knight. However, Marisa did say that Honey knew the answers to Remilia's questions. Honey is omniscient and knows everything, including the answers to Remilia's questions, so Marisa's statement is true. Since she is not an insane knave, yet she utters true statements, she must be a sane knight._

—/—

**Alice's Army of Fairy Dolls**

"So, Alice's Army of Dolls are on the loose, you say?" began Remilia.

"Indeed," replied Marisa, "They're not exactly causing trouble, but they are very cleverly and aggressively defending their master's house."

Marisa treated Remilia and her entourage to mushroom tea and mushroom soup. It was a lot better than it sounded, Remilia had to grudgingly admit. Maybe it had some bizarre hallucinogen in it that bamboozled her into believing it to be good, which would go a long way towards explaining why Marisa ate them. She took another sip of the slightly thick tea.

"And what have you found out about them?" Remilia asked.

Marisa ticked off the points as she went through them. "First, that Alice's dolls can change their lying and truth-telling behavior and their beliefs, depending on the day and the week. The dolls come in four types, which I've labeled K, N, 1 and 2. You'll understand why later. They also come in four classes, labeled by yours truly, S, I, A, and B. Again, you'll understand why later." Remilia frowned at the glossing over of those details, but said nothing. After all, Marisa said she would understand later, and Marisa was a sane knight. Marisa continued, "Each doll, whatever other characteristics they have, belong to exactly one of these classes and exactly one of these types."

"So, a doll can be of designation A2, or SK, or IN?" Remilia asked. Marisa nodded in the affirmative, and Remilia smiled a bit, satisfied with the explanation. "But what do the designations mean?"

"That's what I'm about to explain next. At any one time, each doll is in one of three states, which I call 'stages': 1, 2, or 3. In some of these stages, which I call 'truthful stages,' the doll is in what I call a 'truthful mood,' where she never makes a statement or answers a question contrary to her knowledge and beliefs. In the remainder of these stages, she always makes statements and answers questions contrary to her knowledge and beliefs - she's in a 'deceitful mood,' and the stage called a 'deceitful stage.' A K-type doll is never in a deceitful mood, only in a truthful mood, while an N-type doll is never in a truthful mood, only in a deceitful mood."

"I get it. 'K' and 'N' stand for 'knight' and 'knave,'" Remilia summarized.

Marisa touched her nose as she pointed to her, indicating that she had caught on. "Type 1 dolls, however, have stage 1 as their deceitful stage, while the remaining two stages are truthful, and dolls of type 2 have stage 2 as their deceitful stage, with the remaining two stages truthful."

"I guess that the classes refer to the behavior of sanity in the dolls."

"Correct," replied Marisa. "At any one time, a doll is in one of three other states, which I call 'phases': A, B, and C. Some will be 'lucid phases,' where all the doll's beliefs are correct, and others will be 'deranged phases,' where all the doll's beliefs are incorrect."

Remilia picked up the obvious from there. "If I had to guess, class S dolls are always lucid, class I dolls are always deranged, class A dolls are deranged only in phase A and lucid otherwise, and class B dolls are deranged only in phase B and lucid otherwise."

"Give the vampire an explodin' cigar!" shouted the Black-White jovially. She sobered quickly, adding, "Two more rules: First, a doll believes S in her deranged phase if and only if she believes not-S in her lucid phase."

Remilia interrupted Marisa at that point before she could continue to the next rule. "Then changing phases preserves opinionation. If a doll is opinionated on some S while in some phase, then they are opinionated on that same S in any other phase," concluded the vampire.

"Right. Second rule, each doll is completely opinionated on various attributes of both themselves and their sister, including their types and classes, their states and phases both past and current, and whether they are currently reliable or unreliable."

"'Completely opinionated'? 'Reliable'? 'Unreliable'?" Remilia quizzed. More new terms.

"'Reliable' and 'unreliable' are my terms for a doll that currently makes true statements and currently makes false statements respectively, whether they are actually truthful, lucid, or anything else — it's helpful in keeping track of the mental gymnastics," answered the magician.

Remilia carefully sipped up a spoonful of mushroom soup, noting its crude but hearty taste as she contemplated the new term. "Actually, that wouldn't be a bad general term for other kinds of beings, too," she said aloud, noting that several internal reasonings could have been made shorter by the use of the terminology. A second later, she cursed, "Crap. That probably means it'll go into general use, now that I've said it."

Marisa didn't press that observation. "'Completely opinionated on S' is a bit more difficult to define, but its a statement about what I call belief chains based on that statement S. A direct belief in S and a direct belief in not-S are both belief chains based on S. But if we have an X that we know to be a belief chain based on S, then a direct belief in X or belief in non-X is also a belief chain based on S. If you are completely opinionated on some S, then you are opinionated on all belief chains based on S."

Remilia thought about that for a moment. "So if I am completely opinionated on my reliability, then I am opinionated on my belief that I believe that I am not reliable, and also on any chain of belief and negation as long as it has something to do with my reliability."

"Yep."

"Leaving that aside, that these types and classes apply differently to differing stages and phases strongly suggest that a doll will change both at some time or another," Remilia stated aloud, having a strong suspicion where this was going.

Marisa did not disappoint. "Yep. A doll's stage goes from stage 1 to stage 2 to stage 3 then back to 1, changing stage every twenty-four hours at midnight. A doll's phase goes from phase A to phase B to phase C and then back to A, changing phase every week when the sun rises on Sunday. Together, the combination of a doll's stage and her phase is her state: B2 is a state, as is A1 and C3."

Remilia narrowed her eyes. Something about Marisa's testimony did not add up. "Wait a second," she said in confusion, "the current incident hasn't been going on for weeks and weeks."

"Maybe not the incident proper, but Alice's current project started months before. It was quite innocuous and actually kind a' charming, but then this knight/knave, sane/insane thing hit us and it was clear that Alice's dolls were a prototype. Wasn't so charming after that."

"I see."

"Back to the dolls." said Marisa, a little loudly, to steer the discussion back on track. "You know how Alice makes dolls according to standardized doll lines?" The witch noted Remilia's nod. "Each line has a sister line, and each doll in a line is paired with a doll from the sister line. The dolls are made two at a time, one from each sister line, and form a 'matched pair,' and the dolls in a matched pair are called 'sisters.' The dolls of a matched pair will always be in the same stage and same phase as each other, but this is not necessarily true for a doll of the same or their sister line but not from the same matched pair.

"In order to pass these dolls, you haveta solve a specific puzzle they will pose to you. If they do not pose a challenge to you immediately, then you get to ask questions until they do. The only exception are Shanghai and Hourai. Also, when they are encountered outside of Alice's house, they are either single, or paired with their sister."

Marisa and Remilia talked about specific lines of dolls at that point, which Remilia filed away in her mind for later. They will remain unimportant until the correct time.

Marisa wrapped up the explanation, "As you get deeper into the forest, the puzzles get harder. If you don't solve each dolls' puzzle, she won't let you pass."

"Sounds like a lot of trouble. I think this is a time to break out the _danmaku_ and take care of them the old fashioned way."

"Don't," the ordinary magician warned. "If you try to get past them without answering their challenges, and **especially** if you use _danmaku_ on them, all of Alice's dolls will converge on you... and when I say 'all of Alice's dolls,' I do mean _all_ of them... and they will _wreck your shit!_"

Remilia regarded Marisa's state of health, as did Sakuya. "Is that how you ended up in such a sorry state, Marisa Kirisame?" asked the maid, who was obviously expecting an answer of 'yes.'

"Well... yeah."

"Huh, if they waxed a Forth Stage like yourself, they'll _certainly_ be trouble for us," Remilia snarked lightly, then taking a sip of her mushroom tea.

"Uh, Remi. I'm actually _Extra Stage_ certified, y' know."

Mushroom tea emerged in a fine mist as Remilia spat it out on reflex. Coughing a few times to clear the remainder, Remilia slammed her hands onto the table as she leaned over it with an intense look of disbelief on her face. "You're an Extra Stage Boss!" she shouted.

"Yeah. Had the cert' for years. Alice's Extra Stage, too, an' I've had to face it. Reimu holds Sub-Phantasm certification," said the blonde human magician calmly.

"But... but... you were Forth Stage during the Stolen Moon Disaster!" Remilia sputtered.

"Well, you know ya don't _have_ to play at your rated stage all the time!" Marisa said, a smile finally appearing on her face in enjoyment of seeing Remilia's charisma breaking before her eyes. "Yuyuko, a Stage Six at least, played at Stage One during that Mausoleum incident just a few weeks ago."

Whatever charisma Remilia had lay in shambles at her feet. Marisa, a mere human witch and a thief to boot, was rated at a higher stage than her, the Queen of the Night! It was quite shocking to the vampire. After a moment of barely restrained weeping, she wheeled about and pointed at Sakuya, who retained her composure. "SAKUYA! MAKE A NOTE!" she shrieked, "As soon as this incident is over, I'm training for Extra Stage certification!"

"It will not be done, m'lady," Sakuya said.

Marisa stared at Sakuya with narrowed, suspicious eyes. A crafty smile creeped to her lips. "Sakuya, are you sane?" she asked, out of the blue.

"No," Sakuya answered, then immediately winced in realization.

"Ah. Yer a knave," Marisa said matter-of-factly, grinning at Sakuya's unwitting reveal. "My condolences."

_79. The Locked Box of Marisa Kirisame_

Marisa rose from her place and weaved around the piles of junk that littered her house outside this small little 'living' area. She took up a small box with a brass-detailed keyhole on its front face. "One more thing: this strong-box was left on my doorstep this morning, with instructions that I was to give it to you," she said as she set it down before Remilia. "The note said it was a problem about keys."

Remilia took it and applied a gentle pull to its top. It rattled, but refused to open.

Sakuya glanced at the locket and hazarded a guess. "M'lady, it is not probable at all that the keys you have received from the fairies will be useful for opening this locked box."

"Be careful, Lady Scarlet. You have to use a specific key, and turn that correct key in a specific direction — clockwise or counterclockwise, or the lock will permanently jam," warned Honey Humblebee, the omniscient fairy.

Remilia huffed, merely laying both keys out on the table with a clatter: ward-lock keys cast in gold or brass, one with a diamond shaped bow, the other with a heart shaped bow. But which one should she use, and which way should she turn?

"You'll notice that bolted to the top of the box are four plates of differing metals —gold, silver, copper, and iron— and each engraved with the chemical symbol of their elements and three statements," Honey continued. Remilia did not need to look at it to see that it was so, Honey being a sane knight, but she looked anyway. As expected, it was so. "Each plate has been inscribed in its entirety by either the left or right hand of a single skilled kappa, Eta the Kappa, who is a hermit living in this forest. That is to say, for example, if the kappa's right hand has engraved a single inscription in the copper plate, then the kappa's left hand has not engraved any inscription in the copper plate."

"Is this kappa sane or insane?" asked Remilia.

"That's for you to decide, along with whether Eta is right- or left-handed, or indeed a he or a she. I assure you that Eta is one or the other of those," answered the omniscient fairy.

Remilia looked over the inscriptions at this point.

On the gold plate was inscribed the following:

**~Au~**

**(1) Gretta Julibee is an outsider fairy and claimed to be either sane or insane, and also claimed that either the diamond-bowed key or the heart-bowed key is the correct key for this lock.**

**(2) No more than two of these plates have been inscribed by my right hand.**

**(3) I believe that the diamond-bowed key is the correct key only if the correct key must be turned counterclockwise to open the lock.**

For the silver plate:

**~Ag~**

**(1) I am female.**

**(2) I am right-handed, and I am insane.**

**(3) Gretta Julibee claimed that the diamond-bowed key is the correct key for this lock.**

For the copper plate:

**~Cu~**

**(1) Actually, I am male, or I am right-handed.**

**(2) Inserting the heart-bowed key and turning it clockwise will jam the lock.**

**(3) This plate was inscribed with my right hand.**

For the iron plate:

**~Fe~**

**(1) If I like green tea, then I am insane.**

**(2) Gretta Julibee claimed to be sane.**

**(3) You will arrive at Alice's home and when you get to Alice's home, cake will be served to you, in your honor.**

Remilia clucked her tongue. "Typical. At least the inscriptions bode well for the kappa race," she said as she chose a key and twisted it in the lock.

_Solution: The diamond-bowed key is the correct key for the lock, and it must be turned clockwise._

_First, let (X N) denote the N'th inscription on the plate of elemental symbol X, so the third inscription on the gold plate is (Au 3)._

_(Ag 1) and (Cu 1) cannot both be false, because if they were, then Eta is neither male nor female, and Honey said Eta was one or the other. The only way both inscriptions (Ag 1) and (Cu 1) can be true is if Eta really is female and right-handed. By (Ag 2) Eta must also be insane. This means that Eta's left hand is reliable, and inscribed both plates, while Eta's other hand is not. But (Cu 3) claims that it was inscribed by Eta's right hand, which cannot be because Eta's right hand would be unreliable and inscribe only false inscriptions. The assumption that (Ag 1) and (Cu 1) are both true leads to a contradiction. One must be false._

_Suppose the silver plate has true inscriptions. Then the copper plate has false inscriptions. By (Cu 1), Eta is neither male nor right-handed, and thus must be left-handed. But this contradicts (Ag 2), which asserts (if it is true) that Eta is indeed right-handed. Therefore, the silver plate is not true, and from (Cu 1), we can infer that the inscriptions on it are true (because Eta is not female, but male — contrary to the false inscription (Ag 1)). We can also infer from (Cu 3) that the copper plate has indeed been inscribed by his right hand, and the silver plate by his left._

_(Ag 2) is not true, which means that Eta is either left-handed or sane. Suppose he's left-handed. Then by (Cu 3) Eta is insane, because a left-handed person can only write truth with their right hand if he is insane. Suppose however that Eta is right-handed. Then by (Ag 2), he is sane. Therefore, Eta is sane if and only if he is right-handed._

_We already know the silver plate has only false inscriptions, so if the gold plate bears false inscriptions too, then there are only two plates left that could possibly bear true inscriptions. But this is what inscription (Au 2) implies: that at most two of the plates bear true inscriptions. Therefore, if the gold plate bears only false inscriptions, then one of the inscriptions the gold plate bears is true — a contradiction. Therefore, the gold plate must bear only true inscriptions. Since it is true that at most two of the plates bear only true inscriptions, and the gold and copper plates are those two plates, the iron plate must bear only false inscriptions._

_All inscriptions on the iron plate are false, so the conditional statement (Fe 1) is false too. The only way a conditional can be false is if the antecedent is true, and the consequent is false. Therefore, the consequent that Eta is insane is false. Eta is sane. Also, the antecedent is true, so Eta likes green tea... though that's not important for this problem. Eta is therefore a sane male kappa, which bodes well for the kappa race._

_The copper plate has true inscriptions, therefore by (Cu 2) inserting the heart-bowed key and turning the it clockwise really will jam the lock. Therefore, it cannot be the case that the heart-bowed key is the correct key and the correct key must be turned clockwise are both true. Either the key must be turned counterclockwise, or the heart-bowed key is the incorrect key._

_The gold plate has only true inscriptions, so by (Au 3) Eta really does believe that the diamond-bowed key is the correct key only if the correct key must be turned counterclockwise. We've already established that Eta is sane, so this belief is correct. If the correct key must be turned clockwise to open the lock, then the diamond-bowed key cannot be the correct key; the heart-bowed key must be the correct key. But this cannot be, for by (Cu 2) the heart-bowed key jams the lock if turned clockwise. Therefore, the correct key must be turned counterclockwise._

_Inscription (Au 1) tells us that Gretta Julibee is an outsider fairy, claimed to be either sane or insane, and claimed that either the diamond-bowed or the heart-bowed key was the correct key to the lock. The inscription is on the gold plate, which has no false inscriptions. Therefore, Gretta is indeed an outsider fairy, and as such she is sane. She has also claimed to be either sane or insane, and separately named one of our two keys as the correct key. However, inscription (Fe 2) claims that Gretta has claimed to be sane, but being on the iron plate, it is false. Therefore, Gretta has claimed to be insane, which makes her a knave._

_Inscription (Ag 3) claims that Gretta claimed that the heart-bowed key is the correct key to the lock, but since the inscription is on the silver plate, it is false: by elimination with (Au 1), Gretta actually claimed that the heart-bowed key is the correct key to this lock. But Gretta is a sane knave, therefore, the heart-bowed key is not the correct key; the diamond-bowed key is, and it must be turned counterclockwise._

_Finally, (Fe 3) claims that Remilia will arrive at Alice's home, whereupon they will receive cake, but all inscriptions on the iron plates are false. Therefore, either Remilia will not make it to Alice's house, or if she does arrive, there will be no cake served._

_Yes, it does mean the cake is a lie. I'm very, very sorry._

—/—

The lock released obediently, as Remilia expected, and the vampire opened —in a somehow smug— way the wooden box that contained the secret. The interior of the box was lined with green felt, with two cavities that exactly matched two further keys, again cast in gold or brass, one with a spade shaped bow, the other with a clubs shaped bow — like card suits. Both keys had a black gem embedded in their bows, perhaps an onyx or a jet. Remilia frowned at these contents.

"Oh, bother. More keys," she murmured. "At least with any luck, we won't need to bother with the two old ones." She unrolled and read the note that came along with the keys. "Curse that troll," growled the loli vampire as she finished the note, "It says the keys we already have 'might be useful' in future. So much for luck. Come, Sakuya, Lily, and Honey."

Marisa hobbled up to her doorway, and leaning on the door frame, called out, "Don't forget what I tol' ya about them dolls!"

"Yeah, yeah."

—/—

**Interlude: Deeper in the Forest of Insanity**

"Come, Remilia Scarlet... Sakuya Izayoi..." hissed a female figure cast in shadow, "Come into the glorious bliss that is insanity!" She let out a long, loud peal of cackling.

On the other side of the room, a doll dressed in red shook a thin, wide metal sheet, making a crashing sound that rolled across the room. Two dolls dressed in blue and wearing safety goggles attended a flask where a measure of magnesium burned brightly, as a third rapidly opened and shut a blind that selectively permitted and blocked that magnesium-originated illumination in a flashing staccato upon the mysterious figure.

_(To be continued... unless I'm INSANE!)_

* * *

Notes: I split this chapter up, owing to the ever burgeoning length and the fact that the last part of it is having gelling problems. The conclusion to Remilia's journey through the Forest of Magic will be soon.


	7. Remilia Faces the Seven Colored Magician

_Touhou Project by Shanghai Team Alice_

_Knights and knaves accredited to Raymond Smullyan_

* * *

In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.

All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond Smullyan.

Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.

Remilia Scarlet, the Scarlet Devil, has awoken to find herself unable to lie, and therefore a knight. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.

It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.

Remilia and Sakuya, accompanied by Lily White and Honey Humblebee, are making their way towards the abode of Alice Margatroid to assay her type and sanity, as well as find out what she was up to. The abode is guarded by rings of dolls, which Marisa warned not to challenge with _danmaku_ (citing her copious injuries).

The dolls had the peculiar property of shifting their lying habits depending on which day it was, and their sanities depending on the week. Each doll cycled through three stages, 1, 2, and 3, progressing through these stages one per day. Each doll also cycled through three phases, A, B, and C, progressing through these phases one per week. Each doll's stage and phase is synchronized with a counterpart from a sister line, the doll's 'sister.' When Alice's dolls travel, they are always singly or paired with their sister.

Each doll had a type that dictated how the doll will behave during that day. A type K doll (K for 'knight') is always 'truthful;' they speak according to their beliefs. In contrast, a type N doll (N for 'knave') is always 'deceitful;' they speak contrary to their beliefs. If the doll's type was 1, the doll is deceitful for as long as they remain in stage 1, and truthful otherwise; similarly, type 2 dolls are deceitful in stage 2, and truthful otherwise.

Each doll also had a class that dictated their sanity for that week. A class S doll (S for 'sane') is always 'lucid;' every belief they have is true. A class I doll (I for 'insane') is always 'deranged;' every one of their beliefs is wrong. Class A dolls are deranged in phase A and lucid otherwise, and similarly class B dolls are deranged in phase B and lucid otherwise.

Each doll is completely opinionated on various attributes of both themselves and their sister, including their types and classes, their states and phases both past and current, and whether they are currently reliable or unreliable, thus they are opinionated on all belief chains based on those attributes. Furthermore, a doll that is opinionated about some subject while lucid will remain opinionated on that subject while deranged.

Marisa had also given her particular information about each line of dolls. The party will use that information to navigate each ring of dolls as they come to them. While Remilia did not know what to expect at Alice's house, she did know that there will be no cake served once she arrived.

* * *

**Knights and Knaves:**

**A Logical Adventure of Gensokyo**

by Wyrm

_Chapter 7:_

_In Which Remilia Scarlet Confronts the Seven Colored Magician_

* * *

**Dolls from the Siberian Wastes ~ Tibet and Russia**

_"Tibet and Russia are the first sister lines. Tibet line dolls dress in a traditional tibetan style, while Russia line dolls dress —surprise, surprise— in traditional russian dress. All Tibet dolls have a designation of S1, and all Russia dolls, S2."_

These were the words of Marisa Kirisame concerned with the Tibet and Russia line of Alice's dolls. "This is the first time we will be encountering a being that actually changes the way they lie or tell truth," Remilia said aloud, even though she knew Sakuya had probably deduced this.

A sign was posted beside the path, loudly advertising that they were entering the domain of Alice's dolls. "Come on, my friends," Remilia said as she and Sakuya strolled forward, followed by the hovering Lily and Honey, entering hostile territory.

_80a. Together with Tibet and Russia_

It didn't take long for them to encounter a pair of these dolls, one Tibet doll and one Russia doll. According to Marisa's information, this would be a matched pair.

Remilia recalled that Tibet and Russia were always lucid, and together in one of three stages, 1, 2, or 3, that would progress by one stage every day. Tibet (being type 1) would lie (be deceitful) in stage 1, and tell the truth on other days. Russia (being type 2) would lie in stage 2, and tell the truth at other times.

Without any prompting, Tibet and Russia made their statements.

"Yesterday I was in a deceitful mood," said Tibet.

"Yesterday I was in a deceitful mood too," added Russia.

Tibet then issued their challenge: "Will either of us be deceitful tomorrow, and if so, who?"

Remilia smiled. It wasn't that difficult a challenge, but then, it was only the first ring.

_Solution: If Tibet is reliable today, she is telling the truth and was deceitful yesterday. The only stage where this is true is stage 2. If Tibet is unreliable today, she was not deceitful yesterday. The only stage where this is true is stage 1. So the dolls' current stage is 1 or 2._

_If Russia is reliable today, then she was deceitful yesterday. The only stage where this is true is stage 3. If Russia is unreliable today, then she was actually truthful yesterday. The only stage where this is true is stage 2. So the current stage is 2 or 3._

_Thus, the current stage can only be 2 for this doll pair. Tomorrow, they will be in stage 3, so neither doll will be deceitful tomorrow._

_80b. A Lone Tibetan Doll_

A while later, the group encountered another Tibet doll. She was alone; if her sister was about, she was not near.

"I was in a deceitful mood yesterday," she began, and added, "I will be in a deceitful mood again tomorrow." She then issued her challenge: "What stage am I currently in?"

Remilia solved this one easily, too.

_Solution: Her statements are either both true or both false. If she's telling the truth both times, then she is in a deceitful stage two days in a three day cycle, which a doll of type 1 does not have. Therefore, she is deceitful today, and since she is type 1, she must be in stage 1._

_80c. Solitary Russian Doll_

A few minutes later, they encountered another of the dolls. This time, it was a Russia doll. There was no guarantee that this doll was the sister to the Tibet doll they encountered previously.

"If I was in a deceitful mood yesterday, then I will be in a deceitful mood again tomorrow. What stage am I in?" it challenged.

Remilia did not find it that much of a challenge.

_Solution: If Russia was in a deceitful mood yesterday, then her stage has advanced since yesterday and she is in a truthful mood today. This means that the implication is true, and the antecedent is true, which implies the consequent is true: that she will be in a deceitful mood tomorrow. But tomorrow the doll's stage will have advanced twice from her deceitful stage, and her next deceitful mood is on the day after that! Therefore, yesterday she was not in a deceitful mood._

_This makes the antecedent false, and as such the entire implication is true (an implication with a false antecedent is always true). Therefore, today she is still not in a deceitful mood, and because yesterday she was also not in a deceitful mood, then tomorrow she will be in a deceitful mood (as her deceitful moods come in a three day cycle). Since Russia is a type 2 design, tomorrow she will be in stage 2. Therefore, today she is in stage 1._

—/—

**Italian Dolls from the Cities ~ Napels and Turin**

_"Napels and Turin are the next sister lines. They dress identically in sea green Florentine dresses, and the dolls themselves have black hair both. They are indistinguishable, so don't bother identifying them visually. Napels has the designation S1, and Turin, S2. They're usually alone."_

Marisa's words meant that Napels and Turin basically followed the same rules as Tibet and Russia, only this time they were indistinguishable from each other.

_81a. An Italian Doll by Herself_

The first time they encountered one of these dolls, she was alone. Remembering Marisa's warning, she could have been a Napels or Turin doll.

"I am in a deceitful mood today, or I am Napels," the doll announced.

Remilia thought about it. She could deduce one thing about the doll already, one vital datum required for unharassed passage, but she could not figure out the other. "Is your sister deceitful today?" she asked.

The doll answered, "Yes." Then the doll made her challenge: "Will either of us be deceitful tomorrow?"

_Solution: If the doll is in a deceitful mood today, then her statement is false, and she is neither Napels nor in a deceitful mood. But this is a contradiction, so the doll cannot be in a deceitful mood; she must be in a truthful mood. And since she isn't in a deceitful mood, she must be Napels._

_Napels is a type 1 doll, and her sister is type 2. Since she is in a truthful mood today, her answer to Remilia's question of whether her sister is in a deceitful mood is the truth: her sister Turin is in a deceitful mood, which means the dolls are in stage 2. Tomorrow, they will be in stage 3, and thus neither will be deceitful._

_81b. Another Italian Doll by Herself_

It was not long before they encountered another of these dolls. She looked identical to the last doll, Italian dress and all. "I am in a deceitful mood today, and I am Turin," she announced. "What stage am I?"

"At least this one is fairly easy," remarked Remilia before answering the doll's challenge.

_Solution: If the doll is in a truthful mood, then she is certainly not in a deceitful mood, which means that her statement is a lie. But this is a contradiction, so she must be in a deceitful mood today. Since she is in a deceitful mood, she cannot be Turin (otherwise her statement is true), so she must be Napels. Since Napels is a type 1 doll, and this Napels doll is in a deceitful mood today, she must be in stage 1._

—/—

**Draw Me Like One of Your French Dolls ~ Nancy and Nice**

_"Nancy and Nice are yet another pair of sister lines. They are virtually indistinguishable, save for their dresses, which are colored blue and green, both in french fashion. However, they often exchange garbs, so don't depend on them. Also, while I've figured out that one doll is designated S1, and the other S2, I have no clue which doll has which designation."_

Remilia had remarked at that point that the dolls were named after cities or countries, either in Europe, or in Asia. Marisa expressed the fact that he didn't particularly care about that. As to the dolls themselves, they were like Napels and Turin, except that the designations were unknown.

_82a. The First Conversation with Nancy and Nice_

Two dolls in matching green and blue french dress rose to block the group, Nancy and Nice. Remilia decided to designate the doll dressed in blue, 'Doll B,' and the doll dressed in green, 'Doll G.'

Doll B said, "If I'm Nancy, then she's Nice."

In her turn, Doll G said, "If she's Nice, then I'm Nancy."

Doll B issued the challenge: "What is our current stage?"

Although Remilia could not tell which doll was Nancy and which was Nice, she was able to name their stage, and so answer their spoken challenge.

_Solution: If Doll B is Nancy, then Doll G is Nice; and if Doll B is Nice, then Doll G is Nancy. This is obvious from the fact that dolls are paired up. However, this is precisely what the dolls assert, so the both of the dolls are in a truthful mood. The only stage where both dolls are truthful is stage 3. It is impossible to tell who was who or their individual designations, however._

_82b. All One Ever Wanted to Know about Nancy and Nice_

Remilia spotted the second pair of Nancy and Nice dolls before they could drop off their perch on a high branch to confront them. "Hello," greeted the immature vampire. The two dolls nodded in greeting, then descended to confront the four travelers. In her mind, Remilia again labeled the green-wearing doll 'Doll G' and the blue-wearing doll 'Doll B.'

Doll B began, "Today we are not in stage 3."

Doll G followed, "In fact, today we are in stage 2."

Doll B gave the final statement. "Tomorrow, Nancy will be in a deceitful mood."

Doll G issued the challenge: "Figure out (1) the lines of each of us, (2) what stage we are in, and (3) our types."

Remilia grinned. Those three statements were quite sufficient to answer the green doll's challenge.

_Solution: It is impossible for the two dolls to be in stage 3, because if they were, Doll B could not lie and say that they were not in stage 3. Therefore, they are in stage 1 or 2. Doll B is therefore telling the truth — they are indeed not in stage 3. Thus, Doll G is deceitful, so they are not in stage 2. This leaves them in stage 1, Doll B is type 2 and Doll G is type 1._

_Since Doll B tells the truth today, Nancy will indeed be in a deceitful mood tomorrow. By tomorrow, however, the dolls will have transitioned to stage 2, which means Doll B will be in a deceitful mood tomorrow, while Doll G will be back in a truthful mood. Therefore Doll B is Nancy, who is an S2 doll and Doll G is Nice, an S1 doll._

—/—

**Country Dolls of the West ~ France and Holland**

_"Then we have the sister lines, France and Holland. These two lines are easily distinguishable, with France wearing a flowing green frilly french dress, while Holland wears a black netherlands dress with a white apron. Of course, while they may be encountered separately, they only associate with their sister. They are the first lines to have differing classes as well as types, with France an S1 and Holland a BN."_

This time the rules were different. France's class of S indicated that she was always lucid (all her beliefs are true), while Holland's class of B meant that she was deranged (all her beliefs are false) during phase B in a three phase cycle, A, B, and C, which progressed to the next phase each week, and lucid for all other weeks. France's type of 1 meant she was deceitful in stage 1, but truthful otherwise; Holland's type of N meant that she was always deceitful.

Two dolls of this description descended before Remilia and her companions. Both were blondes, Remilia idly noted.

_83. Lying Dolls_

Holland said to Remilia, "I am unreliable today, but I was reliable this day last week."

France added, "I had lied today."

Holland then issued the challenge: "What is our stage and phase?"

_Solution: Holland cannot be reliable today, because if she were, then her statement is false which it cannot be if she were reliable. Therefore, she is unreliable, and her statement is false. But she is indeed unreliable today, so therefore it must not be true that she was reliable last week — she must have been unreliable last week too. Therefore, there has been two weeks in a row when Holland has been unreliable, which only occurs when she is lucid, so she will be reliable next week, when she will be deranged. Holland is class B, so next week they will be in phase B, so this week they are in phase A._

_France's statement is more subtle. If France is deceitful today, then her statement seems to imply that she has not lied today, despite the fact that she is deceitful today. Yet if she has not lied today, then she is not deceitful — a straight paradox with no way out. However, this only applies if her statement refers to a whole day of not lying, and it doesn't. The statement is in the past tense, so the lies she refers to were before the time that she made her statement. Her statement can be refined to: "I had told a lie today prior to this statement."_

_Of course, if this is the truth, then she had told a lie today and therefore must be in a deceptive mood, given that she is of designation S1. But then she cannot tell the truth and say that she had lied today, so this statement must be a lie; she has not lied today before that statement. But that doesn't mean she has told the truth today, either. This is her first statement today. So she is deceitful today and the dolls are therefore in stage 1._

—/—

**Dolls of Old China**

_"The sister lines, Beijing and Huizhou, are much like France and Holland, respectively, except the two lines are indistinguishable from each other. They wear chinese cheongsams, yellow and white, and often exchange them like Nancy and Nice. Like the other dolls, they only associate with their sister. Beijing is of designation S1, and Huizhou, BN."_

These words echoed in Remilia's head as two dolls in chinese cheongsams dropped down to confront them, clad in yellow and white. Remilia labeled the doll in the yellow cheongsam 'Doll Y,' and the doll in the white cheongsam 'Doll W.'

_84. China Dolls_

"I am class B," announced Doll Y.

"I am type 1," claimed Doll W.

Doll Y continued, "I believe I am type N."

Doll W then issued the challenge: "Which of us is Beijing, and which of us is Huizhou, and what is our current stage?"

Remilia snorted. To be sure, it was more intricate this time, this challenge, but it still fell to her reasoning.

_Solution: From her first statement, if Doll Y is Beijing, then she is currently deceitful, because Beijing is actually class S. On the other hand, if Doll Y is Huizhou, then she is currently deranged, because Huizhou really is of class B, but is also type N, so she can only be reliable if she is in phase B. Either way, Doll Y is in a deceitful mood._

_By Doll W's statement, if Doll W is Beijing, then she is currently truthful, because Beijing really is of type 1 and, being class S, is always lucid. If Doll W is Huizhou, then she is currently lucid, because she actually is of type N and always deceitful, and only unreliable when she is lucid. Therefore, Doll W is lucid._

_Because Doll Y is currently in a deceitful mood, her second statement implies that she is not of type N. For because she is deceitful, she doesn't really believe that she believes she is of type N. Because she is completely opinionated about her type, she believes that she doesn't believe she is of type N._

_Suppose she is lucid. Then that belief is correct, and she doesn't believe she is of type N. Again, this implies she believes she is not of type N, and again, because she is lucid, she really isn't of type N. Suppose instead she is deranged. Then that belief is incorrect, and she actually does believe that she is of type N, and because she is deranged, she isn't of type N. Therefore, Doll Y isn't of type N. She must be of type 1, and furthermore, is Beijing, and Doll W is Huizhou, and because Beijing is currently deceitful, the dolls are in stage 1._

—/—

**Twin Cities on Opposite Sides of the World ~ Kyoto and London**

_"Let's see... France/London... Beijing/Huizhou... ah, Kyouto and London! These dolls are distinguishable, with Kyotos wearing pink kimono, while Londons wear red dresses with white trim and black bearskin hats. One is class S, and the other is class A, and one is type 1 and the other is type 2. I don't know which doll is which, though."_

Remilia was amused that she had to consult a cheat sheet for that one. She didn't even need one to recall that a class of A meant that she was deranged in phase A. Of course, every doll had the same cycles of phases and stages

_85. The Lion and the Chrysanthemum_

A kimonoed doll dropped out of the canopy, obviously a Kyoto. She announced, "I am class A."

The doll in red, London, announced as she settled behind Kyoto, "I believe I am unreliable."

Kyoto added, "London was deceitful yesterday."

London issued the challenge then: "What are our designations, and what are our phase and stage?"

Remilia passed this challenge.

_Solution: If London is currently truthful, then she really is unreliable, because a doll that is both truthful and lucid cannot say "I believe I am unreliable." For if she were truthful and lucid, then then she really does believe she is unreliable, but being truthful and lucid, she is actually reliable, and so believes something that is not true — assuming lucidity leads to a contradiction. Therefore, if she is truthful, she must be deranged. On the other hand, if she is currently deceitful, then she doesn't really believe that she is unreliable, and because she is opinionated on her reliability, she believes she is reliable. But the only way she can be reliable and deceitful is if she is deranged, so we have a deranged doll that correctly believes she is reliable, which is impossible. Therefore, she cannot be deceitful. She is truthful, and therefore deranged._

_Because London is deranged, she is not class S, but class A. By the same token, the dolls are both in phase A. Also because London is class A, Kyoto is class S and therefore lucid. Kyoto is also unreliable today, as she claimed to be of class A, so she must be in her deceitful stage today._

_Kyoto is the deceitful one today, therefore London was not deceitful yesterday and is not today. Therefore, London will be deceitful tomorrow. Since Kyoto is deceitful today, she will not be deceitful tomorrow. Because at least one of them is deceitful today and tomorrow, the day after they will be in stage 3, so they are currently in stage 1. Since Kyoto is currently deceitful, she is type 1 and London is type 2._

—/—

**The Favored Dolls of Alice Margatroid**

_"Finally, we have Shanghai and Hourai, Alice's two favorite dolls. Only one of each of these dolls exist, and of course are sisters to each other. Their types are 1 and 2, and their classes are A and B, but I do not know which doll has which type and which class. Even if I did know, I still wouldn't know which doll is which because they have basically identical body-forms. Unlike the others, their challenge is implicit: to deduce everything about them."_

Remilia leaped back as two shapes shot down to raise a cloud of dust with a thunderous slam. "So, we meet again, Shanghai and Hourai," the vampire remarked, seeing the two familiar figures just barely visible through the screen of dust.

Then the two dolls arose, dressed in simple blue Anne of Green Gables type dresses, with white aprons and red ribbons tying back their long, waist length hair. As Remilia had guessed, they were Shanghai and Hourai, Alice's two favorite dolls, and as promised, completely indistinguishable visually.

_86. Shanghai and Hourai_

The only difference between the two Remilia could see was that one was wearing a pleasant, friendly smile, while the other was wearing an annoyed frown. Remilia mentally and uncreatively labeled the smiling doll 'Doll S' and the frowning doll 'Doll F.'

"What's with the smiling and frowning?" asked the vampire.

The smiling doll —Doll S— said, "Shanghai always smiles and Hourai always frowns."

The frowning doll —Doll F— threw her sister a sharp look. "That's not true!" she protested.

Doll S ignored Doll F's protest. "This day last week, I was deranged."

"You just currently and falsely believe that. I was the one who was deranged last week," countered the frowning doll.

"No, it is _you_ who currently believes false things!" Doll S replied hotly, keeping her smile.

"Enough," Remilia broke in, halting their argument. She then asked of the frowning doll, "Were you reliable this day last week?"

"Yes, I was," she answered.

Remilia tapped her chin. She was able to decipher most of what she needed to know, but there was still a few things she could not figure out from their statements so far.

"You, the one with the frown. Will Hourai smile if I ask her to?"

"If you say 'please,'" Doll F said gruffly.

"Hourai, will you please do so?"

What Remilia saw next enabled her to figure everything out.

_Solution: In their first pair of statements, the dolls contradict each other, so they cannot both be reliable or both unreliable. Therefore, they are not both lucid and truthful — one must be either deranged or deceitful. By the same token, it cannot be that one is lucid and truthful and the other is deranged and deceitful, because then they'd both be reliable, nor can they be one lucid but deceitful and the other deranged but truthful, because then they'd both be unreliable._

_If Doll S is currently reliable, then she is Shanghai and Doll F is Hourai. For if Doll S is reliable, then Shanghai indeed always smiles and Hourai always frowns. Since Doll S is indeed smiling, she must be Shanghai, and not Hourai who always frowns, and Doll F (who is frowning) Hourai. This fact implies that if Doll F is Hourai and she always frowns, then she is currently unreliable._

_Both dolls claim to have been deranged last week, and one being unreliable and the other not, exactly one of them really was deranged last week. Since the dolls are of class A and B, one of the dolls was only in a deranged phase last week, so last week the dolls could have only been in phases A or B, not C. Furthermore, each doll accuse the other of being currently deranged, and again exactly one doll is reliable in her claims, so one really is deranged at the moment. This can only happen in phases A or B, and they cannot be in phase A because that would imply last week they were in phase C, which cannot be the case. Therefore, this week the dolls are both in phase B._

_Suppose Doll S is unreliable. Then she was not deranged last week. This makes Doll F reliable, so it is true that the smiling doll only believes she was deranged last week — Doll F blames her sister's current unreliable state on her derangement. The smiling doll then seriously believes that she was deranged last week and is therefore truthful, but deranged. Suppose, on the other hand, the smiling doll is reliable. Then she was indeed deranged last week. This means that this week, the smiling doll is not currently deranged. Since she is also reliable, she is also currently truthful. In either case, the smiling doll is in a truthful mood. By the same token, the frowning doll is also in a truthful mood. Because they are both in a stage that is truthful, the dolls are currently in stage 3. Combined with the previous derivation, the dolls are currently in state B3._

_Doll F claims to have been reliable that day one week ago. Today, the dolls are in state B3. Last week this same day, they must have been in state A2, because that day one week ago was seven days ago, which means that their stages have advanced seven times to leave them in stage 3, which means they must have been in stage 2 that day, and their phase has advanced one time to arrive at phase B, so last week they were in phase A. If Doll F is currently reliable, she was deranged and reliable that day last week, which implies that she was deranged and deceitful — she is a doll of designation A2. If Doll F is currently unreliable, then last week she was unreliable too, which means that she was neither deranged nor reliable, which means she was lucid and deceitful — she is a doll of designation B2. Either way, she is a doll of type 2, so Doll S is of type 1._

_Remilia was able to figure out all of this before asking Hourai to smile. She could not tell which one was Hourai, nor whether which of the dolls was class A and which was class B._

_If Doll S is Hourai, then Remilia's asking Hourai to smile would have no effect. For we already know that if Doll S is Hourai, she is unreliable, which makes Doll F (Shanghai) reliable. This means her statement about Hourai smiling if Remilia asks her politely is true, but of course the consequent is already true, so Hourai (the smiling doll) will continue to smile._

_If Doll F is Hourai and she is unreliable, then it is not true that she will smile if Remilia asks her politely — if Remilia asks politely, then Hourai must continue to frown (if she smiles, she is reliable). Since Remilia has asked politely, Hourai would remain frowning._

_Either way, if Hourai is the smiling doll, or if she is the frowning doll and unreliable, she will not change her state of smiling or frowning, and Remilia will be unable to tell which doll is Hourai, and bring her no closer to figuring out who is who or what their designations are._

_But Remilia was able to figure out everything. Therefore, someone changed their state of smiling or frowning. Hourai cannot be either the smiling doll (because she would continue to smile), nor unreliable (because she would continue to frown), so Hourai must be Doll F, and is now smiling._

_This proves that Hourai is Doll F, and that she is reliable. It also proves that Shanghai is unreliable. Since they are both in their truthful state, Hourai (Doll F) is lucid and Shanghai (Doll S) is deranged. Thus, the dolls designations are B1 for Shanghai, and A2 for Hourai._

—/—

\title The Seven Colored Magician

"You did well, Lady Remilia Scarlet," Hourai said, curtseying in midair while still smiling the smile that formed on her lips just seconds before. "Do you have keys with bows shaped like card suits?" In answer, Remilia produced her keys. Hourai nodded at this practical demonstration. Even though Remilia's word was just as good, it seemed to give her a lot more satisfaction to present to the doll material proof. "Two of those keys unlock the front and back doors of Alice's house, one for each door. They must be turned either clockwise or counterclockwise to open the door. One way will unlock the door, but the other way will jam the lock. The direction they must be turned are not necessarily the same."

_"Orleans and Lillie are the next-to-last sister lines, the last being Shanghai and Hourai, but more on those two later. Orleans and Lillie are probably the toughest of the dolls to make sense of, because they are completely indistinguishable. They can also be of any designation - and that's not just because I don't know either: every one of their lines run the entire gamut of designations. This is usually of no consequence because the only thing they do is guard Alice's house, and make sure that you not only have the bronze keys with bows shaped like card suits, but also that you use the right key on the right lock. However, I do know that no pair of sisters have matching type or class."_

"You know, we haven't met an Orleans or a Lillie doll yet," said Remilia aloud.

"That's because the only pair about is currently guarding Alice's house," Hourai replied. The doll saw their curiosity piqued, and continued, "They know everything there is to know about Alice's locks —which keys to use on each lock, and which direction they must be turned— at least when they're lucid."

Alice's doll took a moment to scrutinize Remilia and company, scratching her chin out of seeming habit. "Yes, I think you'll do nicely," she said, mostly to herself. Then she started long the path again, bade the four to follow. "Come. I shall lead you to Alice's home. With me as your escort, you will encounter no more challenges from other doll patrols."

_87. Alice's Doors_

As promised, with Hourai as escort, the troupe was unharassed as they made their way to the Margatroid manor. Remilia could catch glimpses of fairy dolls peeking out briefly before scurrying away. Soon, they came onto the clearing where Alice's house lay. As promised, a pair of Orleans and Lille dolls floated before the front entrance, armed and armored, and with a serious look on their faces.

"I feel that there is no trap."

Remilia nodded at Sakuya's words. "Assuming you haven't gone insane already, we should be cautious," Remilia agreed to the implied truth behind Sakuya's lie. The vampire turned to Hourai. "Should we approach from the front or the back door?" she asked.

"Alice is waiting behind one of these doors. If you open the wrong door, then Alice will put a stake through your heart."

Remilia swallowed. A stake through her heart wouldn't _kill_ her, but it would incapacitate her, allowing the mad magician to do other unspeakable things to her.

Honey fluttered in front of Remilia, blocking her progress temporarily. She added an important point to her advice, "I must caution you also that this will be quite a puzzle to solve. I suggest taking notes."

Remilia felt a touch of annoyance at the fairy's sudden imposition, but relented. "Right. Sakuya, your notebook, and use your truthful hand," she commanded her maid.

The maid flipped out a pad of paper and a pen, holding it in her left hand. Her truthful hand. "I shall not do so, m'lady," lied the maid.

Remilia gave her servant a nod, then stepped forward.

The two dolls immediately reacted, crossing their lances in front of her, as if daring her to cross swords with them. Remilia's only reaction was to hold up her four keys. The dolls were suspicious, but were at ease. "State your questions, stranger," ordered the doll wearing blue.

Remilia ignored the harsh tone of the doll. "First, please tell me a bit about yourselves," she started.

"We are not both currently unreliable."

Sakuya quickly jotted down the statement of the doll bedecked in red finery. Remilia took a moment to peek over Sakuya's shoulder, to see that she had helpfully labeled Red's first statement as (Red-1). She nodded at the red doll to continue.

"In addition, my sister is neither class S nor class A."

Remilia saw this statement was labeled (Red-2).

"Finally, Orleans is wearing blue," the red doll finished.

Remilia guessed this would be labeled (Red-3). A quick check confirmed this.

The blue-clothed doll then began to speak. "I am not both lucid and deceitful. I am also not of type 2," she began.

Remilia peeked over Sakuya's shoulder again, and saw those statements labeled (Blue-1) and (Blue-2) respectively. Remilia noted that not a single word was left off by her maid's scribbling. No shorthand. The words were written as they were spoken. Remilia felt momentarily proud.

"As to our classes, Orleans is of class S and I am class B. One of us is not of type K. As to our current phase, we are now in phase B," added the blue-dressed doll.

Remilia checked to see that these three statements were labeled (Blue-3), (Blue-4) and (Blue-5) respectively.

Remilia waited a few beats for any other forthcoming statement, but the dolls remained silent. It seemed they had stated all they wanted to say about each other. Time for her to move on. "Okay, what can you tell me of the locks?"

Again, the red doll spoke first. "Either you have used one of the correct keys before on a previous lock, or the correct keys have the same jewel set in their bows."

_(Red-4),_ thought Remilia. By this time she trusted Sakuya could keep up with the transcription.

"The keys must be turned the same way in their respective locks."

_(Red-5)._

The blue doll broke in this time. "I believe that turning the spade-bowed key counterclockwise will not jam at least one of the locks."

_(Blue-6),_ thought Remilia, still keeping count of the statements in her head.

The red doll followed up with a quite similar statement. "I believe that turning the spade-bowed key clockwise will jam both locks."

_(Red-6),_ thought the vampire girl.

The blue one spoke again. "If you had asked me this day last week, I could have claimed that the correct key for the back turned counterclockwise. If you ask me again this day next week, I could claim that the correct key for the back must be turned clockwise."

_(Blue-7) and (Blue-8)._

"If the front door lock takes a red-jeweled key, then the back door lock also takes a red-jeweled key," came Red's final statement on the keys and locks of Alice's abode, which Remilia knew would be labeled (Red-7).

Again, the silence was unbroken for several beats, indicating the dolls would say nothing more on the subject. The vampire glanced towards her maid, who was furiously scribbling away on the notepad, trying to commit the dolls' words to paper. Her brow was glistening with sweat, and her mouth was pursed tight. After she finished with the transcription, she sighed and wrang out her left arm. While that arm may be as skillful and legible as her dominant arm, it wasn't actually used to lots of writing.

Remilia decided to wrap this up. "I know that Alice will stab me through the heart if I open the 'wrong door.' Please tell me which door is the correct door."

"We are both reliable today if and only if the front door is the door that is not 'wrong,'" said the red one.

Remilia glanced towards Sakuya, and saw her complete transcription of the statement that the vampire knew to be (Red-8), if she got her statement count right. Her left arm was now looking very pained, and Remilia held up her hand. "I think we'll be fine without any more transcription, Sakuya." Turning towards the dolls, she asked one last question, "Red one, what class do you believe yourself to be?"

"I believe I am class I," came the doll's answer.

Remilia took the notebook off Sakuya's hands. It was written in her maid's neat handwriting, the writing of this knightish hand indistinguishable from the handwriting of her actual dominant hand. She scanned the notes, trying to fit the logic together. It took a few minutes of working through the logic, but she was finally able to truthfully announce, "Orleans, Lille, I thank you. I have figured out, amongst other things, which key goes to which lock, which direction they should be turned in, and which entrance to use."

_Solution: The first step is to establish the reliability and identity of both Red and Blue._

_If Red is reliable, it's certainly true that they are not both unreliable (Red is reliable), and thus, her statement (Red-1) is the truth. Also, Red really isn't Orleans, because Red is not wearing blue; her sister is (Red-3). On the other hand, if Red is unreliable, then those two statements imply that it is not true that not both dolls are unreliable, so they are both unreliable, and also that Red actually is Orleans because the untruth of her statement implies that Orleans is not wearing blue, so she must be wearing red. Therefore, if Red is unreliable, Blue is as well; and Red is Orleans if and only if she is unreliable._

_If Blue is reliable, then she is not Orleans and is therefore Lille, because she claims Orleans and her are of differing class (Blue-3). But this means that Red is unreliable, because she claimed that Orleans is wearing blue, and Red's unreliability implies Blue's as well. So we have reached a contradiction: if Blue is reliable, she is not reliable. Therefore, Blue is unreliable. Since Blue is unreliable, she is both lucid and deceitful, and of type 2 (Blue-1 & Blue-2). Therefore, she is not class I. This means she can be class S, A, or B, and of type 2. Blue disclaimed that one of them is type K (Blue-4), so we know that one of the dolls is indeed of type K. Since we know Blue is type 2, we know Red is of type K, and is always truthful (though not always reliable)._

_Red is truthful, so her answer to Remilia's last question reveals that she believes that she believes she is of class I. Suppose Red really does believe she is of class I. Then she is reliable, because she has stated something that is true. Furthermore, she believes something that is true by the same token, so she is lucid today: she really does believe she is class I. Consequently, because she is lucid and believes she is class I, she really is of class I. But this cannot be, because class I dolls are always deranged. So if she really does believe that she is of class I, then she is both lucid and deranged at the same time, which is impossible._

_She cannot believe that she is class I, contrary to her statement, so she is unreliable. But we have already shown she is truthful, so she must be deranged. But she is completely opinionated about her class, so if she doesn't believe she is class I, she must believe she is not of class I. But this is false because she is deranged. Therefore, she must be of class I._

_In addition, Red's unreliability leads us to the conclusions that Red is Orleans, Blue is Lille, and they are both unreliable. We've established Red and Blue's identities and reliabilities, and can now move on to the keys._

_Since Orleans is not reliable, then it is not true that one of the correct keys has been used in a previous lock, nor is it true that the correct keys have the same jewel set in their bows, contrary to her statement (Red-4). Therefore, there is one correct key that has a red jewel, and one correct key that has a black jewel. Furthermore, the correct key that has the red jewel cannot be the diamond-bowed key (which was used in a previous lock), so it must be the heart-bowed key._

_Orleans and Lille know everything about Alice's locks while they are lucid. In particular, they know (when lucid) whether or not the spade-bowed key will jam which locks if turned clockwise or counterclockwise. This makes them opinionated on these positions. When deranged, they are still opinionated on these direct statements of fact._

_Orleans is truthful, so her statement (Red-6) means she believes that she believes that turning the spade-bowed key clockwise will jam both locks. But she is deranged, so she in fact does not believe this direct fact. However, she is opinionated on it, so she believes that this direct fact is not true, and again because she is deranged, this direct fact is indeed true: turning the spade-bowed key clockwise in either lock will jam that lock. (Note that if Orleans were lucid, the same result would hold: turning the spade-bowed key clockwise in either lock will jam that lock.)_

_Lille is deceitful, so her statement (Blue-6) means that she believes that she does not believe that turning the spade-bowed key in at least one of locks will not jam it if turned counterclockwise. Since she is lucid, she really does not believe this direct fact, and since she is opinionated on that direct fact, it implies that she believes this direct fact is not true. Again, because she is lucid, the direct fact actually is not true: the spade-bowed key will jam both locks if turned counterclockwise. (Note that if Lille were deranged, the same result would hold: turning the spade-bowed key counterclockwise in either lock will jam that lock.) Thus, the spade-bowed key will jam the lock if turned clockwise or counterclockwise: the spade-bowed key does not fit in either lock and may be ignored hereon. Therefore, the club-bowed key is a correct key._

_Lille is now in stage 2, because she is a type 2 doll and currently deceitful. Last week this day, Lille was in stage 1 (seven days in the past), so she had been truthful that day. She will be truthful again a week from now (seven days hence) because she will then be in stage 3. She is unreliable today, so actually, she could not have claimed that the correct key for the back would be turned clockwise and next week she could not have claimed that same key would need to be turned counterclockwise (Blue-7 and Blue-8). Since Lille is truthful at both these points, the difference must be because she is lucid at one of these points and deranged at the other point. Either last week she was deranged, or next week she will be deranged (she is currently lucid). Orleans' unreliable statement (Red-2) claiming that Lille is neither class S or A implies that, actually, Lille is either class S or A, but if she is of class S, then Lille cannot change from lucid to deranged. This means she must be of class A._

_Lille's statement (Blue-5) implies the dolls aren't really in phase B this week. Lille is lucid this week and she is of class A, so they are not in phase A this week. They must therefore be in phase C, and next week they will be in phase A. Lille will be deranged that week, so next week this day, she will be truthful and deranged, so she will be unreliable. Therefore, the fact that she could not claim that the correct key for the back should be turned counterclockwise is because that would be the truth. So the key to the back door must be turned counterclockwise. By Orleans' unreliability and her statement (Red-5), the keys must be turned opposite ways. Since the back door key must be turned counterclockwise to open it, the front door key must be turned clockwise._

_Orleans statement (Red-7) is false, so the implication is false, so its antecedent is true while its consequent is false. This means that the front door takes a red-jeweled key and the back door takes a black-jeweled key. The only red-jeweled key that can be a correct key is the heart-bowed key, while the only black-jeweled key in the running is the club-bowed key. Similarly the whole logical equivalence of statement (Red-8) is false, and so the two are both reliable today if and only if the front door is not the door that is not wrong. However, because Orleans (and indeed Lille) is unreliable today, the antecedent is also false, so it must be the case that the front door is not the wrong door._

—/—

Remilia unlocked and threw open the front door, confidently.

***THUNK***

"M'lady!" shouted Sakuya in surprise, shock and dispair.

Remilia was just then feeling the shock of something impacting her breastbone, even as her brain processed the fact that a dark shape had lunged at her as soon as she opened the door. Remilia could only look down and dumbly note that a wooden stake was protruding from her breast, probably piercing her heart, driven firmly and all at once into her sternum by the deceptively willowy arms of Alice Margatroid. Alice's face bore a look of crafty and slightly sneering satisfaction.

The Seven Colored Magician had staked her, she realized, and had been waiting for her at the front door.

"What?" she said, her voice wavering as the strength drained from her, "But... I thought... Hourai..."

"She did," replied Alice, "But she _didn't_ say that if you chose the correct door, that I would _not_ stake you through the heart."

"Bugger," the staked vampire managed to croak out before darkness claimed her.

—/—

Remilia awoke to find herself seated in a chair, bound by the preternaturally strong filaments of Alice's puppet strings, puppet strings held taut by quintuplet of Orleans and Lille dolls. She was not sure if they were the same dolls as were guarding Alice's house when she and Sakuya arrived and she was unsure of how much time had passed since she had been staked.

Speaking of Sakuya, she was across the room from her, in heavy wooden stocks and chained securely by tight manacles on her wrists and legs. Even if she were to activate her time-manipulation powers, it would avail her naught — the manacles would hold her just as securely in her timeless space as they do in this temporal space.

"Sakuya!" shouted Remilia. The Perfect, Elegant Maid lifted her head to face her mistress, and Remilia saw that she was less than perfect, less than elegant. It almost seemed as if several days had passed.

"Mistress... You have revived before today..." groaned the maid.

Remilia immediately deciphered her lie. It strongly implied that she had spent at least a day, perhaps several, unconscious. "Sakuya! Are you well? What did Alice do to you?"

"I am well-fed, or well-rested," Sakuya said weakly, her face slightly gaunt and wearied. "I have been given an inadequate amount of potable water, though. As to the rest, as of now, she has done something other than give me water."

"Is this still the same day, Sakuya?" Remilia asked, glancing at the light of the setting sun.

"I do indeed know whether this day is the day we arrived," replied Sakuya, which Remilia took to mean she did not in fact know.

A loud clack of the twisting of a doorknob announced the arrival of their ultimate captor, Alice Margatroid. An unfriendly smile graced her lips, and upon a metal tray she carried three wicked looking syringes filled with sickly-green serums.

Remilia wasn't about to be intimidated by a bunch of syringes, like a little child. "What are those, Seven Colored Magician?" she asked defiantly.

"Something whose significance I will soon demonstrate." With that, Alice turned around and walked out of the room, closing the door behind her. A very brief time past, probably only a minute or so, before she returned to the room, this time carrying the white-bedecked form of Lily White, bound and gagged and struggling like an earthworm plucked out of the ground by a crow. "To each of my victims I pose a challenge. If she answers correctly within a time of fifteen minutes, I set her free, unharmed and without alteration. If she fails to guess, or guesses falsely, I administer this serum." Alice raised Lily White's writhing and distraught form and added, "This child has failed the challenge, and she will suffer the consequences."

Alice laid the struggling fairy upon a small table and strapped her down. She then picked up one of the syringes, and roughly jabbed it into an arm laid bare. Lily mewled in agony through her gag as the gunk was injected into her system. The fairy's eyes unfocused, then glazed over as the serum took its sinister effect. Within minutes, the fairy herald's eyes cleared once more, but Remilia didn't think that this meant that she was all right again.

Alice released the fairy, taking the gag off. "Now, Lily-dear, are you a knight or a knave?" she asked.

Lily smiled and declared, "I'm a knave!"

A cold chill ran up Remilia's spine. She knew what significance that statement had for Lily's sanity. "She's insane now!"

"You are correct," Alice replied. She then launched on a typical B movie villain explanation, though Remilia didn't really know what a B movie villain was, she certainly recognized the cliché. "I have been experimenting with lucidity and derangement these past months before the incident started. The Great Troll commissioned me to do research into knighthood and knavehood with my dolls, as well as these peculiar states. You have encountered the fruits of those experiments. Yukari decided to forego the dimension of lucidity and derangement at large, but it intrigued me to no end. Through my experiments, I developed this serum. It destroys its victim's sanity, flopping all of their beliefs and rendering their victims permanently deranged... insane."

"Like your dolls do as a matter of course, when they switch from lucid to deranged from week to week. The only difference is that the dolls can flip back," Remilia observed.

Alice looked pleased at Remilia's observations. "Correct again, Remilia Scarlet." The Seven Colored Magician looked contemplative for a moment, "I tell you, I had my work cut out for me converting half the forest to insanity in six days."

Remilia felt an even colder chill run down her spine. "Then... all of the insane beings in this forest-"

"Are _my_ doing, yes!" finished Alice Margatroid in triumph.

_88. The Sanity of Alice Margatroid_

"You're mad!" growled Remilia.

"But am I insane?" Alice asked rhetorically, following it with a chuckle. That was the question, wasn't it? _Madness_ and _insanity_ didn't mean the same thing in this forest as they did outside.

Remilia narrowed her eyes. "I know that you're reliable. Otherwise, you would have denied the obvious truth that Hourai had said that you would put a stake through my heart if I chose the wrong door and that she did not say you would put a stake through my heart if I chose the right door. So you do indeed speak truth, regardless of whether you _mean_ to or not," she observed coldly.

"You are quite correct, Remilia Scarlet. I guess all you need to find out is whether I am a knight or a knave, but you won't get that from me. It's too easy," Alice chuckled.

Remilia frowned, puzzling over how to figure out if Alice was insane or not. She was about to probe for more information when she spied Honey Humblebee out the window, waving to her. As her eyes focused on her, she immediately slapped a note on the window, written in big letters (albeit a bit sloppily) reading, _Alice's sanity has been altered exactly once, or I have written this note with my knaved hand! —Honey Humblebee_

Alice heard the noise, and turned around to see what had collided with her window, but the sign was already gone by the time she looked, which lead to Remilia only barely having enough time to remember what was written upon the note herself. Honey made a face in mockery of the magician, then buzzed off.

"Ugh! I've been trying to capture her for ages, ever since my last experiment on her!" Alice huffed, glaring out the window.

"It doesn't matter. I've already figured out you're insane," said Remilia, glaring at Alice.

_Solution: Honey is, of course, sane and omniscient, so she either believes her written statement or its negation, and her belief will be correct. This means Honey's knighted hand is reliable and her knaved hand is unreliable. If she wrote the statement with her knaved hand, then the statement is true, which cannot be because her knaved hand cannot write true statements. It must be written with her knighted hand. But if her note was written with her knighted hand, it certainly wasn't written with her knaved hand, but the statement must be true, so Alice must have had her sanity altered exactly once._

_All the insane beings of the Forest of Magic are the work of Alice Margatroid. Therefore, she must have been sane before her sanity had been altered, and thus insane now._

—/—

"Indeed, I am insane! In fact, I am the _first_ being of the forest to attain the blessed state of insanity," Alice announced boisterously.

"But you started out as sane," Remilia pointed out.

"And I would have remained so if by chance I had not accidentally dosed myself with my own serum. Ah, to be free of sanity's shackles, such a blessed feeling. Now, you and your faithful —if fibbing— maid shall be free of sanity too!"

If Remilia didn't already know Alice was insane, she would have believe she was making stuff up. It was if she was acting in a primary school production taught by Ms. Keine, only Alice was serving up great slabs of ham. She could feed the entire human village off the ham that Alice was serving up.

Nevertheless, Remilia felt the need to counter Alice's nonsense. "You call it 'shackles,' I call it 'grounding.' I've never before been so envious of a doll, you knavish nutter. At least some of them are lucid part of the time."

"Do not trivialize your situation, Remilia Scarlet! Your jest is not funny!" Alice exploded in rage. Remilia silently answered that it wasn't intended to be. "Any more of that, and I will not even give you the usual chance to escape your fates!"

"So you were saying," Remilia said, dropping the previous thread of conversation. "It seems rather sporting and generous of you, Seven Colored and Insane Magician. Why do you do this?"

Alice blinked quizzically. "Does it seem so? I'll have you know that I have not a generous or sporting bone in my body. I derive an enormous sadistic... pleasure... watching my many victim squirm and worry as they flit about the air impotently, and scribble desperately trying to figure out the answer. It is more than fair compensation for the infinitesimal chance that they may solve my challenge."

Remilia well noted Alice's use of 'infinitesimal;' it offered no comfort whatsoever — quite the opposite. "You suppose correctly, Remilia Scarlet: I have not lost a victim yet. I run very little risk," concluded Alice.

"I have no choice anyway. What is your challenge?" asked the bound vampire.

_89. The Challenge of the Mad Magician_

"You see, there is a statement S that has an almost magical property. If there is any other statement X whose truth or falsehood you want to ascertain, all you would have to do is ask someone who is opinionated on X, 'Is S equivalent to X?' If you get 'yes' for an answer, X must be true; if you get 'no', X must be false.

"For instance, 'Is S equivalent to the statement that you are a knight?' and she answers 'yes,' then that person is a knight. If you were to ask her 'Is S equivalent to the proposition that you are sane?' then if she answers 'no,' she is insane. All provided they are opinionated on their knighthood and sanity, of course."

Remilia was very curious about the identity of this sentence. "What is this sentence S?" she asked in genuine curiosity.

Alice's smile tightened, amplifying its nastiness. "Ah, that is for you to deduce! It is your challenge!" she said, bopping Remilia's nose with her finger. Remilia nipped impotently at the digit, but Alice quickly removed it from harm casually. "You have fifteen minutes. You'd better think hard; the stakes are quite high," she stated, stepping back and then through a waiting door.

"Quite high, indeed..." Remilia growled as the door closed behind Alice. So worried for Sakuya's sanity and her own, that Remilia's mind was almost paralyzed with the prospect, and was quite certain that Alice, that sadistic and insane Seven Colored Puppeteer, was watching from her secret hiding place.

Fifteen minutes elapsed, whereupon Alice entered again. There was a wide, sick grin on Alice's face, speaking of her insanity within. She deliberately picked up one of the filled syringes, and moved over to Remilia.

The vampire instinctively tried to get away from Alice, knowing that the syringe contained a substance that would turn her insane, in this special sense, and was repulsed by the idea that she would become a creature that, no matter how normal she seemed on the surface, believed only that which was false. Then, when Alice was almost about to jab the needle into her arm, Remilia's mouth suddenly broke out into a wide grin of realization as she shouted, "Of course! The sentence is..."

It was this sentence that saved her and Sakuya.

_Solution: Many propositions would work, but one possible S is, "You are currently reliable."_

_Suppose the subject is (currently) reliable. If X is true, then the subject's reliability is equivalent to the truth of X, and being reliable, she answers "yes." On the other hand, if X is false, then the subject's reliability is not equivalent to the truth of X, and because she is reliable, she answers "no."_

_Now suppose the subject is (currently) unreliable. If X is true, then the subject's reliability is not equivalent to the truth of X, but being unreliable, she would answer falsely with "yes." On the other hand, if X is false, then the subject's reliability is equivalent to the truth of X, and being unreliable, she would answer falsely with "no."_

_In either case, if X is true, then the subject answers "yes," and if false, the subject answers "no."_

—/—

For a crucial moment, the threads that held Remilia slackened with her shock. The vampire capitalized on this opportunity and lunged at Alice, knocking the syringe into a corner of the room and pinning the _youkai_ magician to the ground.

"Never before has one managed to solve my puzzle and escape with their sanity intact," growled the magician.

"Then it pleases me to be the first," Remilia replied with a sneer. "It could have easily been my fate. I was rather scared out of my mind. But that's over now, Seven Colored Magician." Remilia curled back her lips, exposing her wicked dental apparatus. "Use your dolls to free my maid. Now," she hissed menacingly.

Alice looked almost as if she would not, but relented with a sigh. The Lille and Orleans dolls moved to release Sakuya from the heavy stock and release the manacles holding her fast. Sakuya extracted herself groggily from the stocks, swaying a bit, both tired and drained. Steel returned to her figure, however, when she flicked out a fan of her knives. "On your word, M'lady, I shall not begin to make her pay for her affront!"

"I do not give the word as of yet, Sakuya," Remilia snapped back, keeping her blood-red eyes on Alice. "Anyway, if you somehow managed to capture Honey Humblebee, or another one of a certain class of fairies, they would have been able to answer your challenge immediately," added Remilia after seeing Sakuya move to extract herself from the stocks and manacles that held her. Seeing Alice's confused look, Remilia teased her with a hint, "They're opinionated and sane — think about it, and you may figure out why." Remilia paused, thinking over what she'd just claimed. "Or maybe not. Your beliefs _are_ screwed up at the moment."

"Now that you have me, Remilia Scarlet, what will you do with me?" asked Alice. "Will you give the word to your servant to extract your revenge out of my hide?"

Remilia thought about that for a long moment, staring into Alice's defiant eyes. Then she smiled, deciding against that. After all, Alice was not afraid of her despite her current disadvantage, and Remilia always respected such iron will. "No," the vampire said at last, "Instead, you _will_ say to me that you will not inflict your insanity onto anyone ever again, Alice Margatroid."

Alice narrowed her eyes in annoyance. "Don't make a sane knave out of yourself, Remilia! I believe you don't know for certain that I will make such a statement!"

"That statement works against you, whether you realize it or not. I have seen and manipulated your threads of fate. You _will_ make that statement. I guarantee it," Remilia hissed through a tight smile. "Say it, and be bound by the power of Gensokyo's logic, for you already are."

Alice narrowed her eyes at Remilia. The vampire didn't know what the insane magician was thinking, but she knew Alice's fate. Indeed, she would capitulate right about... _now!_ "Fine. I will not inflict insanity onto anyone ever again," said the Seven Colored Magician with a sigh. She, of course, did not believe that she stated this promise, but it did not matter. She was reliable, and consequently as good as her word. Remilia breathed out a breath she did not realize she was holding.

Just then, one of Alice's favorite dolls, either Shanghai or Hourai, burst into the room, tackling Alice in a hug as she wailed into her bosom. "Master! I'm _so_ happy that you put a stop to this nonsense!" she mewled.

"Hourai?" asked Alice, confirming the doll's identity.

The doll stared up at Alice with dewy eyes. "It isn't right to inflict insanity on people against their will, Master. I'm so glad you said you would stop turning people insane! I turn insane for one week out of every three. I do not find it pleasant when I return to my senses a week after," she babbled.

"While I believe madness is good for you, I shall abide by my words," replied Alice. Hourai giggled and hugged her tighter.

Remilia noted that not only was it true that Alice believed that madness was good for you, and also noted that, being insane, that belief is false. She wondered if she would ever come to appreciate that fact. (Obviously, not until she was sane again.)

"Hourai, how long have we been captured?" Remilia asked.

"It is now morning of the day after," Hourai said.

Remilia knew that Hourai was still reliable due to her quite true statement that she spent every third week deranged, which was true, so she was reliable today. Which means that it was now the next day, as she just said. "Well, it's not as bad as I feared, as we haven't been gone for days and days, but I do think you owe me, my servant, and Lily White breakfast and an apology."

"So be it. I apologize for my mistreatment," Alice said with a bow. "Hourai, please prepare some breakfast for me and our guests."

"Yes, Master!" Hourai said in turn, and fluttered off to prepare some repast.

_90. Curing Spring's Herald_

After a hearty breakfast of pancakes, Remilia was browsing through Alice's stuff for a potential cure for at least Lily White and Letty Whiterock. Alice insisted that she had no general cure, saying that she had no interest in making such a formula. Remilia ignored this, brushing her off easily with a comment that one of her formulas might be able to restore Letty and Lily in particular, even if she did not have a general cure.

Remilia spied a formula that caught her interest. "'Experimental Formula #32: Opinionation Serum.'" She took a vial of the yellow formula out of the felt-padded case it was stored in with several others. "What's this about, Alice?" she asked, presenting the vial.

"A failed formula," Alice said, somewhat morosely, "I was trying to bestow blessed insanity with that serum, but instead induced opinionation. The subjects I used it on turned opinionated instead of insane. Honey Humblebee was one of them."

Remilia turned that fact over in her head, even as she turned over the vial of Opinionation Serum in her hand. Suddenly, her eyes widened in realization. "Alice," she brusquely called to the mad magician, "will this serum cancel out Letty's or Lily's status as seasonal spirits?"

"No," replied Alice.

Remilia smiled. She left the box she extracted the samples from two vials short. She handed one sample off to Sakuya. "Perfect. Sakuya, take this and go find Letty Whiterock. I shall administer this dose to Lily. Lily?"

"Miss Scarlet?" Remilia's vampire speed guided the syringe to Lily, injecting her with the sample before she could yelp in pain, "OW!"

Remilia waited as Lily rubbed her sore bottom, watching her blink and straighten out in wonder, eyes clearing.

"Lily, are you a knight?" asked the vampire.

Lily smiled and shouted in the joy of realization, "YES!"

Remilia smiled. This was indeed a cure to Alice's induced insanity, at least in seasonal spirits.

_Solution: In a previous problem, Lily's statement (prior to being made insane) that seasonal spirits believe that it is exactly one season at any one time implies that such a spirit is opinionated on every season only if she is sane. For if the spirit is insane, then (without loss of generality) she believes the season is currently winter when it is actually currently spring. Because she only believes it is one season at a time, if she believes it is currently winter, she does not believe it is fall. But if she is opinionated on the seasons, if she doesn't believe it is fall, she believes it is not fall. But by hypothesis, it is indeed not fall, but spring! This belief is correct, which cannot be if the seasonal spirit is insane. Therefore, if she is opinionated about the season, she cannot be insane, and must be sane._

_The serum alters seasonal spirits such that they are opinionated without altering the fact that they believe that it is exactly one season at a time. Therefore, if they take the serum, Letty and Lily will continue to believe only one season is the true season, and at the same time be opinionated. If they are opinionated, then they are opinionated about the season, and therefore, by the above reasoning, they will become sane._

—/—

Remilia and Lily found Letty Whiterock milling not far from where they first encountered her, freezing a stream within line of sight with the boulder she had been sitting on. Sakuya had just finished injecting the serum, and pulled the needle from her arm. Soon, Letty's eyes cleared, the serum taking effect.

Lily smiled and threw her arms out wide. "It's good to have you back, Letty!" she said cheerfully.

"It's good to be back, Lily," said Letty calmly, "And I'm glad you're back, too."

"Thanks!"

Remilia could feel the sweetness rotting her teeth. "Okay, okay... enough..." murmured the vampire. "We must go now. I only hope that somebody solves this Incident before the turn of the seasons." With that, Remilia and Sakuya took to the air, heading back to the Scarlet Devil Mansion.

Letty smiled, staring after them as they flew. "You will, Remilia Scarlet. I know it!" she whispered.

—/—

**Interlude: Smullyan's Treatment**

A loud roar of outrage echoed through the Scarlet Devil Mansion. The roar shook the mansion from foundation to roof, and was followed by the quieter but no less audible stream of curses flowing liberally from the mistress's mouth, in a mix of European languages.

The roar startled Flandre out of her concentration, as the clay training sphere she held in her hands popped, her power going out of control and shattering it. "Woah! What's my big sis so steamed about?" asked the blonde loli vampire, suddenly curious.

Patchouli's mouth stretched into a wide, thin smile. "If I had to guess," said the unmoving library magician with a lilting, playful voice, "then she's just discovered Smullyan's treatment of vampires."

_**"WE DO NOT ALL LIE, SMULLYA~~~~~N!"**_

_(To be continued... unless I am a doll!)_

* * *

There will be at most six more chapters, owing to the increasing difficulty of the puzzles I'm constructing, and the fact that I only have so much material. I might have to cut down on the number of puzzles per part to keep them manageable. I hope to never again have a puzzle where one has to keep track of multiple statements.


End file.
